C H A P T E R 9
Topics in Analytic Geometry
Section 9.1
Circles and Parabolas . . . . . . . . . . . . . . . . . . . . 772
Section 9.2
Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . 784
Section 9.3
Hyperbolas . . . . . . . . . . . . . . . . . . . . . . . . . 795
Section 9.4
Rotation and Systems of Quadratic Equations . . . . . . . 807
Section 9.5
Parametric Equations . . . . . . . . . . . . . . . . . . . . 825
Section 9.6
Polar Coordinates . . . . . . . . . . . . . . . . . . . . . . 833
Section 9.7
Graphs of Polar Equations . . . . . . . . . . . . . . . . . 845
Section 9.8
Polar Equations of Conics . . . . . . . . . . . . . . . . . 854
Review Exercises
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Practice Test
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 863
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 886
C H A P T E R 9
Topics in Analytic Geometry
Section 9.1
■
■
■
Circles and Parabolas
A parabola is the set of all points !x, y" that are equidistant from a fixed line (directrix) and a fixed point
(focus) not on the line.
The standard equation of a parabola with vertex !h, k" and
(a) Vertical axis x " h and directrix y " k ! p is
(x ! h)2 " 4p( y ! k), p # 0.
(b) Horizontal axis y " k and directrix x " h ! p is
(y ! k)2 " 4p(x ! h), p # 0.
The tangent line to a parabola at a point P makes equal angles with
(a) the line through P and the focus.
(b) the axis of the parabola.
Vocabulary Check
1. conic section
2. locus
3. circle, center
4. parabola, directrix, focus
5. vertex
6. axis
7. tangent
2. x 2 $ y 2 " !4#2 "
2
2
x 2 $ y 2 " 18
x 2 $ y 2 " 32
3. Radius " #!3 ! 1"2 $ !7 ! 0"2
4. Radius " #$6 ! !!2"%2 $ $!3 ! 4%2
" #4 $ 49 " #53
" #64 $ 49 " #113
!x ! h"2 $ ! y ! k"2 " r 2
!x ! h"2 $ ! y ! k"2 " r 2
!x ! 3"2 $ ! y ! 7"2 " 53
!x ! 6"2 $ ! y $ 3"2 " 113
5. Diameter " 2#7 ⇒ radius " #7
!x ! h"2 $ ! y ! k"2 " r 2
!x ! h"2 $ ! y ! k"2 " r 2
!x $ 3"2 $ ! y $ 1"2 " 7
!x ! 5"2 $ ! y $ 6"2 " 12
7. x 2 $ y 2 " 49
772
6. Diameter " 4#3 ⇒ radius " 2#3
8. x2 $ y2 " 1
9. !x $ 2"2 $ ! y ! 7"2 " 16
Center: !0, 0"
Center: !0, 0"
Center: !!2, 7"
Radius: 7
Radius: 1
Radius: 4
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1. x 2 $ y 2 " !#18 "
Section 9.1
10. !x $ 9"2 $ ! y $ 1"2 " 36
13.
11. !x ! 1"2 $ y 2 " 15
Circles and Parabolas
12. x2 $ ! y $ 12"2 " 24
Center: !!9, !1"
Center: !1, 0"
Center: !0, !12"
Radius: 6
Radius: #15
Radius: #24 " 2#6
1 2 1 2
x $ y "1
4
4
14.
1 2 1 2
x $ y "1
9
9
x2 $ y2 " 4
15.
x2 $ y2 " 9
Center: !0, 0"
Center: !0, 0"
Radius: 2
Radius: 3
4 2 4 2
x $ y "1
3
3
x2 $ y2 "
9 2 9 2
x $ y "1
2
2
#3
2
17. !x 2 ! 2x $ 1" $ ! y 2 $ 6y $ 9" " !9 $ 1 $ 9
!x ! 1"2 $ ! y $ 3"2 " 1
2
x2 $ y2 "
9
Center: !1, !3"
Radius: 1
Center: !0, 0"
Radius:
3
4
Center: !0, 0"
Radius:
16.
773
#2
3
18. !x2 ! 10x $ 25" $ ! y2 ! 6y $ 9" " !25 $ 25 $ 9
19. 4!x 2 $ 3x $ 94 " $ 4! y 2 ! 6y $ 9" " !41 $ 9 $ 36
4!x $ 32 " $ 4! y ! 3"2 " 4
2
!x ! 5"2 $ ! y ! 3"2 " 9
Center: !5, 3"
Radius: 3
Center:
!x $ 32 "2 $ ! y ! 3"2 " 1
!! 32, 3"
Radius: 1
20. 9!x2 $ 6x $ 9" $ 9! y2 ! 4y $ 4" " !17 $ 81 $ 36
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9!x $ 3"2 $ 9! y ! 2"2 " 100
!x $ 3"2 $ ! y ! 2"2 " 100
9
Center: !!3, 2"
Radius:
21.
10
3
x 2 " 16 ! y 2
x 2 $ y 2 " 16
−5
−3 −2 −1
−2
−3
−5
y 2 " 81 ! x 2
y
10
8
6
4
2
x 2 $ y 2 " 81
3
2
1
Center: !0, 0"
Radius: 4
22.
y
5
Center: !0, 0"
1 2 3
5
x
Radius: 9
−10
−6 −4 −2
−4
−6
−8
−10
2 4 6 8 10
x
23.
Chapter 9
Topics in Analytic Geometry
24.
x 2 $ 4x $ y 2 $ 4y ! 1 " 0
!x 2 $ 4x $ 4" $ ! y 2 $ 4y $ 4" " 1 $ 4 $ 4
x 2 ! 6x $ y 2 $ 6y $ 14 " 0
!x 2 ! 6x $ 9" $ ! y 2 $ 6y $ 9" " !14 $ 9 $ 9
!x $ 2"2 $ ! y $ 2"2 " 9
!x ! 3"2 $ ! y $ 3"2 " 4
Center: !!2, !2"
Center: !3, !3"
y
3
2
Radius: 3
−7 −6 −5
y
1
Radius: 2
−3 −2 −1
2 3
−2 −1
x
−6
−7
x 2 ! 14x $ y 2 $ 8y $ 40 " 0
y
!x ! 7"2 $ ! y $ 4"2 " 25
−2
4 6 8 10
14 16 18
x
−4
−6
−8
−10
−12
−14
Center: !7, !4"
Radius: 5
27. x 2 $ 2x $ y 2 ! 35 " 0
x 2 $ 6x $ y 2 ! 12y $ 41 " 0
!x 2 $ 6x $ 9" $ ! y 2 ! 12y $ 36" " !41 $ 9 $ 36
!x 2 $ 2x $ 1" $ y 2 " 35 $ 1
!x $ 3"2 $ ! y ! 6"2 " 4
!x $ 1"2 $ y 2 " 36
Center: !!3, 6"
Center: !!1, 0"
y
9
8
7
6
5
4
3
2
1
Radius: 2
−8 −7 −6 −5 −4 −3 −2 −1
28.
y
10
8
Radius: 6
4
2
−10 −8
−4 −2
2 4 6 8 10
1 2
x
−8
−10
29. y-intercepts: !0 ! 2"2 $ ! y $ 3"2 " 9
x 2 $ ! y 2 $ 10y $ 25" " !9 $ 25
4 $ ! y $ 3"2 " 9
x2 $ ! y 2 $ 5"2 " 16
! y $ 3"2 " 5
Center: !0, !5"
y " !3 ± #5
y
1
−5 −4 −3 −2 −1
−2
−3
−4
−5
−6
−7
−8
x
−4
x 2 $ y 2 $ 10y $ 9 " 0
Radius: 4
x
6
4
2
!x 2 ! 14x $ 4" $ ! y 2 $ 8y $ 16" " !40 $ 49 $ 16
26.
4 5 6 7 8
−2
−3
−4
−5
−6
−7
−8
−9
−2
−3
−4
25.
1 2
1 2 3 4 5
x
!0, !3 ± #5 "
x-intercepts: !x ! 2"2 $ !0 $ 3"2 " 9
!x ! 2"2 " 0
x"2
!2, 0"
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774
Section 9.1
30. y-intercepts: !0 $ 5"2 $ ! y ! 4"2 " 25
Circles and Parabolas
31. y-intercepts: Let x " 0.
! y ! 4" " 0
y 2 ! 6y ! 27 " 0
2
y 2 ! 6y $ 9 " 27 $ 9
y"4
! y ! 3"2 " 36
!0, 4"
y ! 3 " ±6
x-intercepts: !x $ 5"2 $ !0 ! 4"2 " 25
y " 9, !3
!x $ 5"2 $ 16 " 25
!x $ 5"2 " 9
x $ 5 " ±3
!0, 9", !0, !3"
x-intercepts: Let y " 0.
x 2 ! 2x ! 27 " 0
x " !8, !2
x 2 ! 2x $ 1 " 27 $ 1
!!2, 0", !!8, 0"
!x ! 1"2 " 28
x ! 1 " ± #28
x " 1 ± 2#7
!1 ± 2#7, 0"
33. y-intercepts: !0 ! 6"2 $ ! y $ 3"2 " 16
32. y-intercepts: Let x " 0.
! y $ 3"2 " 16 ! 36
y 2 $ 2y $ 9 " 0
No solution
" !20
No y-intercepts
No solution
x-intercepts: Let y " 0.
No y-intercepts
x-intercepts: !x ! 6"2 $ !0 $ 3"2 " 16
x 2 $ 8x $ 9 " 0
!x ! 6"2 " 7
x 2 $ 8x $ 16 " !9 $ 16
!x $ 4"2 " 7
x ! 6 " ± #7
x $ 4 " ± #7
x " 6 ± #7
!6 ± #7, 0"
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x " !4 ± #7
!!4 ± #7, 0"
34. y-intercepts: !0 $ 7"2 $ ! y ! 8"2 " 4
35. (a) Radius: 81; Center: !0, 0"
! y ! 8"2 " 4 ! 49
" !45
x 2 $ y 2 " 812 " 6561
(b) The distance from !60, 45" to !0, 0" is
No solution
#602 $ 452 " #5625 " 75 miles.
No y-intercepts
Yes, you would feel the earthquake.
x-intercepts: !x $ 7"2 $ !0 ! 8"2 " 4
(c)
y
x 2 + y 2 = 812
!x $ 7"2 " 4 ! 64
40
" !60
No solution
− 40
(60, 45)
40
x
− 40
No x-intercepts
You were 81 ! 75 " 6 miles from the outer
boundary.
775
Chapter 9
776
Topics in Analytic Geometry
36. (a) Area " %r 2 " 1800
r2 "
r"
(b) %R 2 " 2400
1800
%
R"
#1800
%
& 27.640 feet
#2400
%
27.640 ! 23.937 & 3.703 longer radius
r & 23.937 feet
37. y 2 " !4x
38. x2 " 2y
Vertex: !0, 0"
Vertex: !0, 0"
Opens to the left since p is
negative.
Matches graph (e).
40. y2 " !12x
p"
1
2
> 0
Opens upward
Matches graph (b).
41. (y ! 1)2 " 4(x ! 3)
39. x2 " !8y
Vertex: !0, 0"
Opens downward since p is
negative.
Matches graph (d).
42. !x $ 3"2 " !2! y ! 1"
Vertex: !0, 0"
Vertex: !3, 1"
Vertex: !!3, 1"
p " !3 < 0
Opens to the right since p is
positive.
p " ! 12 < 0
Opens to the left
Matches graph (f).
43. Vertex: !0, 0" ⇒ h " 0, k " 0
Matches graph (a).
44. Point: !!2, 6"
Graph opens upward.
x " ay2
!2 " a!6"2
x2 " 4py
Point on graph: !3, 6"
1
! 18
"a
1 2
x " ! 18
y
32 " 4p!6"
9 " 24p
Matches graph (c).
45. Vertex: !0, 0" ⇒ h " 0, k " 0
Focus: !0, ! 32 " ⇒ p " ! 32
!x ! h"2 " 4p! y ! k"
x2 " 4!! 32 "y
x2 " !6y
y 2 " !18x
"p
Thus, x2 " 4!38 "y ⇒ y " 23 x2
⇒ x 2 " 32 y.
46. Focus:
! 52, 0"
⇒ p " 52
47. Vertex: !0, 0" ⇒ h " 0, k " 0
48. Focus: !0, 1" ⇒ p " 1
y 2 " 4px " 4! 52 "x
Focus: !!2, 0" ⇒ p " !2
x 2 " 4py " 4!1"y
y 2 " 10x
! y ! k"2 " 4p!x ! h"
x 2 " 4y
y 2 " 4!!2"x
y 2 " !8x
49. Vertex: !0, 0" ⇒ h " 0, k " 0
50. Directrix: y " 3 ⇒ p " !3
51. Vertex: !0, 0" ⇒ h " 0, k " 0
Directrix: y " !1 ⇒ p " 1
x 2 " 4py
Directrix: x " 2 ⇒ p " !2
!x ! h" " 4p! y ! k"
x2
y 2 " 4px
2
!x ! 0"2 " 4!1"! y ! 0"
x 2 " 4y or y " 14 x 2
" !12y
y 2 " !8x
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3
8
Opens downward
Section 9.1
Circles and Parabolas
52. Directrix: x " !3 ⇒ p " 3
y2 " 4px
y2 " 12x
53. Vertex: !0, 0" ⇒ h " 0, k " 0
54. Vertical axis
Passes through !!3, !3"
Horizontal axis and passes through the point !4, 6"
! y ! k" " 4p!x ! h"
x 2 " 4py
! y ! 0"2 " 4p!x ! 0"
!!3"2 " 4p!!3"
2
9 " !12p ⇒ p " ! 34
y 2 " 4px
62 " 4p!4"
x 2 " 4!! 34 "y
36 " 16p ⇒ p " 94
x 2 " !3y
y 2 " 4!94 "x
y 2 " 9x
1
55. y " 2 x 2
56. y " !4x 2
57. y 2 " !6x
x 2 " 2y " 4! 12 "y; p " 12
1
x 2 " ! 14 y " 4!! 16
"y, p " ! 161
y 2 " 4!! 32 "x; p " ! 32
Vertex: !0, 0"
Vertex: !0, 0"
Vertex: !0, 0"
1
Focus: !0, 2 "
1
Focus: !0, ! 16
"
Focus: !! 32, 0"
1
Directrix: y " ! 2
1
Directrix: y " 16
Directrix: x " 32
y
y
y
5
−2
4
−1
1
4
x
2
3
3
2
x
−2
–6 –5 –4 –3 –2 –1
1
–3
–2
2
3
–3
−4
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–1
60. y2 " !x
x 2 " 4!!2"y; p " !2
Vertex: !0, 0"
Focus:
–4
59. x 2 $ 8y " 0
y2 " 4!34 "x; p " 34
!34, 0"
Directrix: x " ! 34
y2 " 4!! 14 "x, p " ! 14
Vertex: !0, 0"
Vertex: !0, 0"
Focus: !0, !2"
Focus: !! 14, 0"
Directrix: y " 2
Directrix: x " 14
y
y
y
6
4
3
4
2
2
−2
x
2
4
6
2
−3
x
58. y2 " 3x
1
−8 −6 −4
−2
−4
−6
−8
−10
4
6
8
x
x
–5
–4
–3
–2
–1
1
–2
–3
777
778
Chapter 9
Topics in Analytic Geometry
61. !x $ 1"2 $ 8! y $ 3" " 0
62. !x ! 5" $ ! y $ 4"2 " 0
! y $ 4"2 " ! !x ! 5" " 4!! 14 "!x ! 5"
!x $ 1"2 " 4!!2"! y $ 3"
Vertex: !5, !4"
h " !1, k " !3, p " !2
Vertex: !!1, !3"
Focus: !5 ! 14, !4" " !19
4 , !4"
y
4
Focus: !!1, !5"
Directrix: x " 21
4
2
Directrix: y " !1
1
x
2
−2
y
−4
−2 −1
−1
−6
−2
−8
−3
−10
−4
−12
−5
1
2
3
4
5
x
6
−6
−7
63. y2 $ 6y $ 8x $ 25 " 0
64. y2 ! 4y ! 4x " 0
! y $ 3"2 " 4!!2"!x $ 2"; p " !2
Vertex: !!2, !3"
! y ! 2"2 " 4!x $ 1"; p " 1
Vertex: !!1, 2"
y
Focus: !!4, !3"
2
x
Directrix: x " 0
–10
–8
–6
–4
–2
y
Focus: !0, 2"
6
Directrix: x " !2
4
–4
x
–6
–4
2
–2
–8
65. !x $ 32 " " 4! y ! 2" ⇒ h " ! 32, k " 2, p " 1
66. !x $ 12 " " 4! y ! 1" ⇒ p " 1
2
2
Vertex: !! 32, 2"
Vertex: !! 12, 1"
Focus: !! 32, 2 $ 1" " !! 32, 3"
Focus: !! 12, 1 $ 1" " !! 12, 2"
Directrix: y " 1
4
Directrix: y " 0
y
y
6
4
5
3
x
−5 −4 −3 −2 −1
1
2
3
−3
−2
x
−1
y " 14!x2 ! 2x $ 5"
10
12
68. 4x ! y2 ! 2y ! 33 " 0
4y ! 4 " !x ! 1"2
y2 $ 2y $ 1 " 4x ! 33 $ 1
y
! y $ 1"2 " 4!1"!x ! 8"
!x ! 1"2 " 4!1"! y ! 1"
6
h " 1, k " 1, p " 1
4
Vertex: !8, !1"
Vertex: !1, 1"
2
Focus: !9, !1"
Focus: !1, 2"
Directrix: y " 0
2
−1
−2
67.
1
x
–2
2
Directrix: x " 7
y
6
4
2
x
4
−2
−4
−6
2
4
6
© Houghton Mifflin Company. All rights reserved.
3
4
Section 9.1
69. x2 $ 4x $ 6y ! 2 " 0
Circles and Parabolas
70. x2 ! 2x $ 8y $ 9 " 0
x2 $ 4x $ 4 " !6y $ 2 $ 4 " !6y $ 6
x2 ! 2x $ 1 " !8y ! 9 $ 1
!x $ 2"2 " !6! y ! 1"
!x ! 1"2 " !8! y $ 1" " 4!!2"! y $ 1"
!x $ 2"2 " 4!! 32 "! y ! 1"
Vertex: !1, !1"
Vertex: !!2, 1"
y
2
Focus: !1,!3"
Focus: !!2, 1 !
3
2
779
" " !!2,
! 12
"
Directrix: y " 52
x
−4 −3 −2 −1
Directrix: y " 1
2 3 4 5 6
−2
−3
−4
−5
−6
−7
−8
y
5
4
3
2
1
−6
−4 −3
x
−1
2
−2
−3
71. y2 $ x $ y " 0
72. y2 ! 4x ! 4 " 0
y2 $ y $ 14 " !x $ 14
y2 " 4x $ 4 " 4!1"!x $ 1"
Vertex: !!1, 0"
! y $ 12 "2 " 4!! 14 "!x ! 14 "
Focus:
!14, ! 12 "
!0, ! 12 "
−4
−3
−2
4
6
8
−6
x
−1
2
−2
−4
1
y1 " ! 12 $ #14 ! x
© Houghton Mifflin Company. All rights reserved.
2
2
To use a graphing
calculator, enter:
y2 "
4
Directrix: x " !2
y
Directrix: x " 12
! 12
6
Focus: !0, 0"
h " 14, k " ! 12, p " ! 14
Vertex:
y
1
−2
!# !x
1
4
73. Vertex: !3, 1",
opens downward
74. Vertex: !5, 3" ⇒ h " 5, k " 3
Passes through: !2, 0", !4, 0"
y " ! !x ! 2"!x ! 4"
"
!x2
$ 6x ! 8
" ! !x ! 3"2 $ 1
!x ! 3"2 " ! ! y ! 1"
Focus: !3,
"
⇒ p"
! y ! k"2 " 4p!x ! h"
! y ! 3"2 " 4p!x ! 5"
1 " 4p!4.5 ! 5"
p " ! 12
3
Focus: !! 2, 0"
1
2
y2
"p
" 4! 12 "!x $ 2"
y2 " 2!x $ 2"
! y ! 3"2 " !2!x ! 5"
76. Vertex: !3, !3" ⇒ h " 3, k " !3
! 94
Passes through: !4.5, 4"
75. Vertex: !!2, 0",
opens to the right
3
4
77. Vertex: !5, 2"
Focus: !3, 2"
!x ! h" " 4p! y ! k"
Horizontal axis: p " 3 ! 5 " !2
!x ! 3"2 " 3! y $ 3"
! y ! 2"2 " 4!!2"!x ! 5"
2
! y ! 2"2 " !8!x ! 5"
x
780
Chapter 9
Topics in Analytic Geometry
78. Vertex: !!1, 2" ⇒ h " !1,
k"2
79. Vertex: !0, 4"
80. Vertex: !!2, 1" ⇒ h " !2,
k"1
Directrix: y " 2
Focus: !!1, 0" ⇒ p " !2
Directrix: x " 1 ⇒ p " !3
Vertical axis
!x ! h"2 " 4p! y ! k"
! y ! k"2 " 4p!x ! h"
p"4!2"2
!x $ 1"2 " 4!!2"! y ! 2"
! y ! 1"2 " 4!!3"!x ! !!2""
!x ! 0"2 " 4!2"! y ! 4"
!x $ 1"2 " !8! y ! 2"
! y ! 1"2 " !12!x $ 2"
x2 " 8! y ! 4"
81. Focus: !2, 2"
82. Focus: !0, 0"
Directrix: x " !2
Directrix: y " 4 ⇒ p " !2
Horizontal axis
Vertex: !0, 2"
Vertex: !0, 2"
x 2 " 4!!2"! y ! 2"
p"2!0"2
x 2 " !8! y ! 2"
! y ! 2"2 " 4!2"!x ! 0"
! y ! 2"2 " 8x
83. y2 ! 8x " 0x
y2
" 8x
84. x2 $ 12y " 0
and
3x
and
x ! $ 2y3 " x $ 2
!y$2"0
y1 " #8x
12y "
y1 "
5
y2 " ! #8x
The point of tangency is
!2, 4".
−6
6
−3
and
x$y!3"0
!x2
y2 " 3 ! x
1 2
! 12
x
The point of tangency is
!6, !3".
4
−4
8
−4
' (
1
1
85. x2 " 2y, !4, 8", p " , focus: 0,
2
2
d1 "
1
!b
2
d2 "
#!4 ! 0" $ '8 ! 12(
2
d1 " d2 ⇒
m"
2
"
17
2
1
17
!b"
⇒ b " !8
2
2
8 ! !!8"
"4
4!0
y " 4x ! 8,
Tangent line
Let y " 0 ⇒ x " 2 ⇒ x-intercept !2, 0".
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Following Example 4, we find the y-intercept !0, b".
Section 9.1
86.
' (
' 2 (y " x
1
p"
2
⇒ p"!
1
2
Focus:
'
Focus: 0, !
'0, 2(
1
#
'
9 1
!!3 ! 0"2 $
!
2 2
(
2
"5
Tangent line: y " !3x !
© Houghton Mifflin Company. All rights reserved.
m"
"
17
8
1
17
$b"
⇒ b"2
8
8
!2 ! 2
"4
!1 ! 0
3
89. R " 375x ! x 2
2
R is a maximum of $23,437.50 when
x " 125 televisions.
25,000
0
'
1
!2 ! 0"2 $ !8 $
8
(
2
65
"
8
1
65
$b"
⇒ b"8
8
8
!8 ! 8
" !8
2!0
y " !8x $ 8
Intercept: !1, 0"
250
0
'
(
1
1
⇒ x-intercept ! , 0 .
2
2
3
'0, ! 18(
d1 " d2 ⇒
2
'! 2, 0(
' (
#
2
9
⇒ 6x $ 2y $ 9 " 0
2
1
1
1
x2 " ! y " 4 ! y ⇒ p " !
2
8
8
d2 "
#!!1 ! 0" $ '!2 $ 18(
Let y " 0 ⇒ x " !
88. y " !2x 2, !2, !8"
1
$b
8
d2 "
y " 4x $ 2
! !9)2" ! !9)2"
m"
" !3
0$3
d1 "
1
$b
8
m"
9
b"!
2
Focus:
(
d1 "
d1 " d2 ⇒
1
!b"5
2
x-intercept:
1
8
1
8
Following Example 4, we find the y-intercept !0, b".
1
d1 " ! b
2
d2 "
781
1
1
87. y " !2x2 ⇒ x2 " ! y " 4 ! y
2
8
2y " x2
4
Circles and Parabolas
Chapter 9
90. (a)
Topics in Analytic Geometry
x2 " 4py
322 " 4p
(b)
' (
1
12
1
x2
"
24 12,288
91. (a) x 2 " 4py, p "
6
3
x2 " 4
y " 6y
2
5
!or y " 6x"
2
4
8 in.
2
512 " x2
1
(b) When x " 4,
x & 22.6 feet
3072 " p
y
'(
12,288
" x2
24
1
1024 " p
3
3
2
− 4 −3 − 2 −1
−1
y"
x2
y"
12,288
(−640, 152)
8
9
p " 36
16 " 4
6
The wire should be inserted
9
4 inches from the bottom.
−4
−2
p " 12,800
19
(6, 4)
2
4
6
19
y " 51,200
x2
x
x
−2
(c)
x
0
200
400
500
600
y
0
14.84
59.38
92.77
133.59
2
94. (a) x2 " 4py passes through point !16, ! 5 ".
95. Vertex: !0, 0%
256 " 4p!! 25 " ⇒ p " !160
y2 " 4px
x2 " 4!!160"y
Point: !1000, 800"
1 2
x2 " !640y or y " ! 640
x
8002 " 4p!1000% ⇒ p " 160
1
(b) !0.1 " ! 640 x 2 ⇒ x " 8 feet
y2 " 4!160"x
y2 " 640x
96. (a) V " 17,500#2 mi)hr & 24,750 mi)hr
(b) p " !4100, !h, k" " !0, 4100"
97. !12.5! y ! 7.125" " !x ! 6.25"2
!12.5y $ 89.0625 " x2 ! 12.5x $ 39.0625
!x ! 0"2 " 4!!4100"! y ! 4100"
! ! 0"x2 " !16,400! y ! 4100"
x
6402 " 4p!152"
(640, 152)
2
−6
4
x 2 " 4py
(b)
y
10
36 " 4p!4"
3
8
inches
3
93. (a)
y
2
16 8
" .
6
3
Depth:
92. x 2 " 4py, !6, 4" on parabola
1
−2
6y " 16
x2 " 4!3072"y
(0, 32 (
y " !0.08x2 $ x $ 4
(a)
10
0
16
0
(b) The highest point is at !6.25, 7.125". The
distance is the x-intercept of &15.69 feet.
© Houghton Mifflin Company. All rights reserved.
782
Section 9.1
98. (a) x2 " !
y"!
1 2
v ! y ! s"
16
16x 2
$s
v2
3
y $ 4 " !x ! 3"
4
4y $ 16 " 3x ! 9
1 2
x $ 75
64
(b) y " 0 " !
783
99. The slope of the line joining !3, !4" and the center
is ! 43. The slope of the tangent line at !3, !4" is 34.
Thus,
16x 2
" ! 2 $ 75
32
"!
Circles and Parabolas
3x ! 4y " 25,
tangent line.
1 2
x $ 75 ⇒ x2 " 75!64"
64
⇒ x & 69.3 ft
100. The slope of the line joining !!5, 12" and the
12
center is ! 5 . The slope of the tangent line
5
at !!5, 12" is 12. Thus,
y ! 12 "
5
!x $ 5"
12
y $ 2#2 "
12y ! 144 " 5x $ 25
5x ! 12y $ 169 " 0,
101. The slope of the line joining !2, !2#2 " and
the center is !!2#2 ")2 " ! #2. The slope of
the tangent line is 1)#2 " #2)2. Thus,
#2
2
!x ! 2"
2y $ 4#2 " #2x ! 2#2
tangent line.
#2x ! 2y " 6#2,
tangent line.
102. The slope of the line joining !!2#5, 2" and the center is 2)!!2#5 " " !1)#5.
The slope of the tangent line is #5. Thus,
y ! 2 " #5!x $ 2#5 "
y ! 2 " #5x $ 10
© Houghton Mifflin Company. All rights reserved.
#5x ! y $ 12 " 0,
tangent line.
103. False. The center is !0, !5".
104. True
105. False. A circle is a conic
section.
106. False. A parabola cannot
intersect its directrix or focus.
107. True
108. False. The directrix y " ! 4
is below the x-axis.
109. Answers will vary. See the reflective property of parabolas, page 599.
110. The graph of x 2 $ y 2 " 0 is a single point, !0, 0".
y
5
4
3
2
1
−5 −4 −3 −2 −1
1 2 3 4 5
x
−2
−3
−4
−5
The plane intersects the double-napped cone at the vertices of the cones.
1
Chapter 9
784
Topics in Analytic Geometry
111. " y # 3#2 ! 6"x " 1#
112. " y " 1#2 ! 2"x # 2#
For the upper half of the parabola,
For the lower half of the parabola,
y # 3 ! !6"x " 1#
y " 1 ! # !2"x # 2#
y ! !6"x " 1# " 3.
y ! #1 # !2"x # 2#.
113. f "x# ! 3x3 # 4x " 2
114. f "x# ! 2x 2 " 3x
Relative maximum: "#0.67, 3.78#
Relative minimum: #1.13 at x ! #0.75
Relative minimum: "0.67, 0.22#
115. f "x# ! x 4 " 2x " 2
116. f "x# ! x 5 # 3x # 1
Relative minimum: "#0.79, 0.81#
Relative minimum: #3.11 at 0.88
Relative maximum: 1.11 at #0.88
Section 9.2
Ellipses
■
An ellipse is the set of all points "x, y# the sum of whose distances from two distinct fixed points (foci)
is constant.
■
The standard equation of an ellipse with center "h, k# and major and minor axes of lengths 2a and 2b is
"x # h#2 " y # k#2
"
! 1 if the major axis is horizontal.
a2
b2
"x # h#2 " y # k#2
(b)
"
! 1 if the major axis is vertical.
b2
a2
■ c2 ! a2 # b2 where c is the distance from the center to a focus.
c
■ The eccentricity of an ellipse is e ! .
a
Vocabulary Check
1. ellipse
2. major axis, center
3. minor axis
4. eccentricity
1.
x2 y 2
"
!1
4
9
2.
x2 y2
" !1
9
4
3.
x2
y2
"
!1
4
25
Center: "0, 0#
Center: "0, 0#
Center: "0, 0#
a ! 3, b ! 2
a ! 3, b ! 2
a ! 5, b ! 2
Vertical major axis
Horizontal major axis
Vertical major axis
Matches graph (b).
Matches graph (c).
Matches graph (d).
© Houghton Mifflin Company. All rights reserved.
(a)
Section 9.2
4.
7.
x2
" y2 ! 1
4
6.
Center: "2, #1#
Center: "#2, #2#
a ! 2, b ! 1
a ! 4, b ! 1
Horizontal major axis
Horizontal major axis
Horizontal major axis
Matches graph (e).
Matches graph (f).
Matches graph (a).
x2
y2
" !1
64
9
Center: "0, 0#
10
8
6
4
2
c ! !64 # 9 ! !55
−10
2 4
10
x
c ! !81 # 16 ! !65
−4
−6
−8
−10
Foci: "± !55, 0#
x2
y2
"
!1
16 81
Center: "0, 0#
y
10
6
4
2
a ! 9, b ! 4,
−4 −2
Vertices: "± 8, 0#
e!
8.
y
−10 −8 −6
Foci: "0, ± !65#
e!
"x # 4#2 " y " 1#2
"
!1
16
25
10.
−2
2
!65
c
!
a
9
"x " 3#2 " y # 2#2
"
!1
12
16
Center: "#3, 2#
a ! 5, b ! 4, c ! 3
a ! 4, b ! 2!3, c ! !16 # 12 ! 2
Vertices: "4, #1 ± 5#; "4, #6#, "4, 4#
Foci: "#3, 2 ± 2#; "#3, 0#, "#3, 4#
Foci: "4, #1 ± 3#; "4, #4#, "4, 2#
Vertices: "#3, 2 ± 4#; "#3, #2#, "#3, 6#
c
3
!
a 5
e!
y
x
−10
Center: "4, #1#
e!
6 8 10
−4
−6
Vertices: "0, ± 9#
!55
c
!
a
8
785
"x " 2#2 " y " 2#2
"
!1
9
4
Center: "0, 0#
a ! 8, b ! 3,
9.
"x # 2#2
" " y " 1#2 ! 1
16
5.
Ellipses
6
c
2 1
! !
a 4 2
y
6
4
4
2
© Houghton Mifflin Company. All rights reserved.
−4 −2
2
−2
6
10
3
x
2
1
−4
−7
−6
−8
11.
−2
"x " 5#2
" " y # 1#2 ! 1
9$4
y
4
Center: "#5, 1#
3
3
a ! , b ! 1, c !
2
%
Foci: #5 "
!5
2
!
2
!5
9
#1!
4
2
&%
, 1 , #5 #
%
& %
!5
2
,1
1
−7 −6 −5 −4 −3 −2 −1
−2
&
&%
−3
−4
& %
3
7
3
13
Vertices: #5 " , 1 ! # , 1 , #5 # , 1 ! # , 1
2
2
2
2
e!
!5$2
3$2
!
!5
3
−4 −3 −2 −1
&
x
1
1
x
Chapter 9
786
12. "x " 2#2 "
Topics in Analytic Geometry
" y " 4#2
!1
1$4
y
x
–3
!3
1
a ! 1, b ! , c ! !a2 # b2 !
2
2
–2
–1
–1
–2
Center: "#2, #4#
Foci:
%
#2 "
!3
2
1
–3
&%
, #4 , #2 #
!3
2
, #4
&
–4
–5
Vertices: "#3, #4#, "#1, #4#
Eccentricity:
!3
2
13. (a) x 2 " 9y 2 ! 36
(c)
y
10
8
6
4
x2
y2
" !1
36
4
(b) a ! 6, b ! 2, c ! !36 # 4 ! !32 ! 4!2
−10 −8
Center: "0, 0#
6 8 10
x
−4
−6
−8
−10
Vertices: "± 6, 0#
Foci: "± 4!2, 0#
e!
c
4!2 2!2
!
!
a
6
3
14. (a) 16x 2 " y 2 ! 16
x2 "
(c)
y
5
4
y2
!1
16
1
(b) a ! 4, b ! 1, c ! !16 # 1 ! !15
−5 −4 −3 −2
2 3 4 5
x
Center: "0, 0#
Foci: "0, ± !15#
e!
15. (a)
!15
c
!
a
4
9x2 " 4y2 " 36x # 24y " 36 ! 0
(c)
y
6
9"x2 " 4x " 4# " 4" y2 # 6y " 9# ! #36 " 36 " 36
4
"x " 2#2 " y # 3#2
"
!1
4
9
(b) a ! 3, b ! 2, c ! !5
Center: "#2, 3#
Foci: "#2, 3 ± !5 #
Vertices: "#2, 6#, "#2, 0#
e!
!5
3
3
2
−6 −5 −4 −3 −2 −1
−2
x
1
2
© Houghton Mifflin Company. All rights reserved.
−4
−5
Vertices: "0, ± 4#
Section 9.2
16. (a) 9"x 2 # 6x " 9# " 4" y 2 " 10y " 25# ! #37 " 81 " 100
(c)
9"x # 3#2 " 4" y " 5#2 ! 144
−2
"x # 3#
" y " 5#
"
!1
16
36
2
y
2
2
(b) a ! 6, b ! 4, c ! !20 ! 2!5
2
4
8
10 12
x
−10
Center: "3, #5#
−12
Foci: "3, #5 ± 2!5#
Vertices: "3, #5 ± 6#; "3, 1#, "3, #11#
e!
2!5 !5
!
6
3
17. (a)
6x2 " 2y2 " 18x # 10y " 2 ! 0
%
6 x2 " 3x "
&
%
(c)
y
&
9
25
27 25
" 2 y2 # 5y "
! #2 "
"
4
4
2
2
%
6 x"
3
2
&
2
"x " 32 #2
4
%
"2 y#
"
5
2
&
2
" y # 52 #2
12
4
2
! 24
−6
x
−4
2
−2
!1
(b) a ! 2!3, b ! 2, c ! 2!2
Center:
Foci:
%# 32, 52&
%# 32, 52 ± 2!2&
Vertices:
© Houghton Mifflin Company. All rights reserved.
e!
!2
%# 32, 52 ± 2!3&
!
!3
!6
3
%
18. (a) "x 2 # 6x " 9# " 4 y 2 " 5y "
%
"x # 3#2 " 4 y "
"y "
"x # 3#
"
36
9
2
&
25
! 2 " 9 " 25
4
5
2
&
#
2
5 2
2
Foci:
%3, # 52&
%3
Vertices:
e!
!3
2
± 3!3, #
5
2
&
%9, # 52&, %#3, # 52&
y
6
5
4
3
2
1
! 36
−3
!1
(b) a ! 6, b ! 3, c ! !36 # 9 ! !27 ! 3!3
Center:
(c)
−1
−2
−3
−4
−6
−7
−8
1 2 3 4 5 6
9 10
x
Ellipses
787
Chapter 9
788
19. (a)
Topics in Analytic Geometry
16x2 " 25y2 # 32x " 50y " 16 ! 0
(c)
y
2
16"x2 # 2x " 1# " 25" y 2 " 2y " 1# ! #16 " 16 " 25
1
(x # 1)2
" (y " 1)2 ! 1
25$16
x
–2
–1
5
3
(b) a ! , b ! 1, c !
4
4
1
3
–2
–3
Center: "1, #1#
Foci:
%4, #1&, %4, #1&
7
Vertices:
e!
20. (a)
1
%4, #1&, %# 4, #1&
9
1
3
5
9x2 " 25y2 # 36x # 50y " 61 ! 0
(c)
y
9"x2 # 4x " 4# " 25" y2 # 2y " 1# ! #61 " 36 " 25
2
9"x # 2#2 " 25" y # 1#2 ! 0
1
(b) Degenerate ellipse with center "2, 1# as the only point
x
1
12x 2 " 20y 2 # 12x " 40y # 37 ! 0
%
12 x 2 # 1 "
&
2
5
" 20" y " 1#2 ! 60
1 2
2
−2
−1
1
2
3
x
−2
−3
" y " 1#
!1
3
2
"
(b) a ! !5, b ! !3, c ! !5 # 3 ! !2
%12, #1&
%12 ± !5, #1&
%12 ± !2, #1&
Eccentricity:
1
−3
"x # #
Foci:
2
&
%
Vertices:
y
1
" 20" y 2 " 2y " 1# ! 37 " 3 " 20
4
1
12 x #
2
Center:
(c)
!2
!10
c
!
!
a !5
5
−4
© Houghton Mifflin Company. All rights reserved.
21. (a)
2
Section 9.2
22. (a)
%
&
4
4
36 x 2 " x "
" 9" y2 # 4y " 4# ! #43 " 16 " 36
3
9
%
36 x "
2
3
&
2
1
4
(c)
Center:
" 9" y # 2#2 ! 9
"x " 23 #2
"
!1 # 14 !
1
(b) a ! 1, b ! , c !
2
36x 2 " 9y 2 " 48x # 36y " 43 ! 0
Foci:
y
!3
2
%# 23, 2 ± 1& ! %# 23, 1&, %# 23, 3&
%# 23, 2 ± 23&
!
Eccentricity:
3
!3
c
!
a
2
1
−2
−1
1
2
x
−1
23. Center: "0, 0#
a ! 4, b ! 2
24. Vertices: "± 2, 0# ⇒ a ! 2
Endpoints of minor axis:
Vertical major axis
y2
x2
"
!1
4
16
%0, ± 2& ⇒ b ! 2
3
x2
y2
" 2!1
2
a
b
x2
y2
"
!1
2
2
"3$2#2
x2 4y2
"
!1
4
9
© Houghton Mifflin Company. All rights reserved.
25. Center: "0, 0#
26. Vertices: "0, ± 8# ⇒ a ! 8
a ! 3, c ! 2 ⇒ b ! !9 # 4 ! !5
Foci: "0, ± 4# ⇒ c ! 4
Horizontal major axis
b2 ! a2 # c2 ! 64 # 16 ! 48
x2 y2
" !1
9
5
Center: "0, 0# ! "h, k#
" y # k#2 "x # h#2
"
!1
a2
b2
x2
y2
"
!1
64 48
27. Center: "0, 0#
28. Center: "0, 0#
c!3
c!2
a ! 4 ⇒ b ! !16 # 9 ! !7
a ! 6 ⇒ b ! !36 # 4 ! !32 ! 4!2
Horizontal major axis
Horizontal major axis
x2
x2
y2
"
!1
36 32
16
"
y2
7
!1
789
%# 23, 2&
Vertices:
" y # 2#2
!1
1
Ellipses
3
790
Chapter 9
Topics in Analytic Geometry
29. Vertices: "0, ± 5# ⇒ a ! 5
30. Vertical major axis
Passes through: "0, 4# and "2, 0#
Center: "0, 0#
a ! 4, b ! 2
Vertical major axis
"x # h#
" y # k#
"
!1
2
b
a2
x2
y2
"
!1
b2 a2
x2
y2
"
!1
b2 25
x2
y2
"
!1
4
16
2
2
Point: "4, 2#
42
22
"
!1
2
b
25
16
4
21
!1#
!
b2
25 25
400 ! 21b2
400
! b2
21
x2
y2
"
!1
400$21 25
21x2
y2
"
!1
400
25
31. Center: "2, 3#
a ! 3, b ! 1
Center: "2, #1# ⇒ h ! 2, k ! #1
Vertical major axis
Endpoints of minor axis: "2, 0#, "2, #2# ⇒ b ! 1
"x # h#2 " y # k#2
"
!1
b2
a2
"x # h#2 " y # k#2
"
!1
a2
b2
"x # 2#2 " y # 3#2
"
!1
1
9
"x # 2#2 " y " 1#2
"
!1
4
1
33. Center: "4, 2#
34. Center: "2, 0#
a ! 4, b ! 1 ⇒ c ! !16 # 1 ! !15
c ! 2, a ! 3 ⇒ b2 ! a2 # c2 ! 9 # 4 ! 5
Horizontal major axis
Horizontal major axis
"x # 4#
" y # 2#
"
!1
16
1
"x # 2#2 y 2
" !1
9
5
2
2
35. Center: "0, 4#
c ! 4, a ! 18 ⇒ b2 ! a2 # c2 ! 324 # 16 ! 308
Vertical major axis
x2
" y # 4#2
"
!1
308
324
© Houghton Mifflin Company. All rights reserved.
32. Vertices: "0, #1#, "4, #1# ⇒ a ! 2
Section 9.2
36. Center: "2, #1# ⇒ h ! 2, k ! #1
Vertex:
37. Vertices: "3, 1#, "3, 9# ⇒ a ! 4
Center: "3, 5#
%2, 2& ⇒ a ! 2
1
3
Minor axis of length 6 ⇒ b ! 3
Minor axis length: 2 ⇒ b ! 1
Vertical major axis
"x # h#
" y # k#
"
!1
b2
a2
"x # h#2 " y # k#2
"
!1
b2
a2
"x # 2#2 " y " 1#2
"
!1
1
"3$2#2
"x # 3#2 " y # 5#2
"
!1
9
16
2
"x # 2#2 "
2
4" y " 1#2
!1
9
39. Center: "0, 4#
38. Center: "3, 2# ! "h, k#
a ! 3c
Vertices: "#4, 4#, "4, 4# ⇒ a ! 4
Foci: "1, 2#, "5, 2# ⇒ c ! 2, a ! 6
a ! 2c ⇒ 4 ! 2c ⇒ c ! 2
b2 ! a2 # c2 ! 36 # 4 ! 32
22 ! 42 # b2 ⇒ b2 ! 12
"x # h#2 " y # k#2
"
!1
a2
b2
Horizontal major axis
"x # h#2 " y # k#2
"
!1
a2
b2
"x # 3#
" y # 2#
"
!1
36
32
2
2
40. Vertices:
"5, 0#, "5, 12# ⇒ a ! 6
x2
" y # 4#2
"
!1
16
12
41.
Endpoints of minor axis:
"0, 6#, "10, 6# ⇒ b ! 5
© Houghton Mifflin Company. All rights reserved.
Center: "5, 6# ⇒ h ! 5, k ! 6
"x # h#2 " y # k#2
"
!1
b2
a2
x2 y 2
" !1
4
9
c ! !9 # 4 ! !5
c ! !36 # 25 ! !11
e!
!5
c
!
a
3
"x 2 # 10x " 25# " 9" y 2 " 4y " 4# ! #52 " 25 " 36
"x # 5#2 " y " 2#2
"
!1
9
1
a ! 3, b ! 1, c ! !9 # 1 ! 2!2
e!
c
2!2
!
a
3
x2
y2
"
!1
25 36
a ! 6, b ! 5,
x 2 " 9y 2 # 10x " 36y " 52 ! 0
"x # 5#2 " 9" y " 2#2 ! 9
42.
a ! 3, b ! 2,
"x # 5#2 " y # 6#2
"
!1
25
36
43.
Ellipses
e!
!11
c
!
a
6
791
Chapter 9
792
44.
Topics in Analytic Geometry
4x 2 " 3y 2 # 8x " 18y " 19 ! 0
4"x 2 # 2x " 1# " 3" y 2 " 6y " 9# ! #19 " 4 " 27
4"x # 1#2 " 3" y " 3#2 ! 12
"x # 1#2 " y " 3#2
"
!1
3
4
a ! 2, b ! !3, c ! !4 # 3 ! 1
e!
c
1
!
a 2
46. Vertices: "0, ± 8# ⇒ a ! 8, h ! 0, k ! 0
45. Vertices: "± 5, 0# ⇒ a ! 5
4
c
4
! ⇒ c! a!4
5 a
5
Eccentricity:
Eccentricity: e !
b2 ! a2 # c2 ! 25 # 16 ! 9
1
c
!
2 a
1
c
!
2 8
Center: "0, 0#
c!4
Horizontal major axis
b2
x2
y2
" !1
25
9
!
a2
#
c2
! 64 # 16 ! 48
x2
y2
" 2!1
2
b
a
x2
y2
"
!1
48 64
(b) Vertices: "± 50, 0# ⇒ a ! 50
y
80
Height at center:
40 ⇒ b ! 40
60
(0, 40)
Horizontal major axis
20
(−50, 0)
−40
(50, 0)
−20
20
40
x
−20
48. (a)
y2
x2
"
! 1, y ≥ 0
2500 1600
(b) a ! 16, b ! 12
y
20
16
x2
y2
"
! 1, y ≥ 0
256 144
(0, 12)
8
(−16, 0) 4
−20
−12 −8 −4
x2
y2
"
!1
a2 b2
(16, 0)
4 8 12 16 20
x
−8
−12
−16
−20
x2
y2
" 2 ! 1 be the equation of the ellipse. Then b ! 2 and
2
a
b
a ! 3 ⇒ c2 ! a2 # b2 ! 9 # 4 ! 5. Thus, the tacks are placed
49. Let
at "± !5, 0#. The string has a length of 2a ! 6 feet.
(c) For x ! 45,
452
y2
"
! 1.
2500 1600
%
y2 ! 1600 1 #
452
2500
&
y2 ! 304
y ' 17.44
The height five feet from
the edge of the tunnel is
approximately 17.44 feet.
(c) When x ! 10,
y 2 ! 144
10
!1 # 256
2
y ' 9.4 > 9.
Hence, the truck will be able
to drive through without
crossing the center line.
© Houghton Mifflin Company. All rights reserved.
47. (a)
Section 9.2
50.
Ellipses
793
51. Area of ellipse ! 2"area of circle#
y
$ab ! 2$ r2
40
$a"10# ! 2$ "10#2
− 20
20
x
40
$a"10# ! 200
a ! 20
−40
Length of major axis: 2a ! 2"20# ! 40 units
%or 23x
x2
y2
" 2!1
2
"97$2#
23
2
!%972&
97
, b ! 23, c !
2
a!
"
2
2
y2
!1
"97$2#2
&
# "23#2 ' 4.7
Distance between foci: 2"4.7# ' 85.4 feet
52. Center: "0, 0#, e ! 0.97
53. a " c ! 4.08
2a ! 35.88 ⇒ a ! 17.94 ⇒ a2 ' 321.84
e!
c2
a # c ! 0.34
2a ! 4.42 ⇒ a ! 2.21 ⇒ c ! 1.87
c
c
⇒ 0.97 !
⇒ c ! 17.4018
a
17.94
!
a2
Ellipse:
#
b2
⇒
x2
321.84
"
b2
!
a2
y2
19.02
#
c2
b 2 ! a 2 # c 2 ⇒ b 2 ! 1.3872
' 19.02
x2
y2
"
!1
4.8841 1.3872
!1
55. For
54. a " c ! 947 " 6378 ! 7325
a # c ! 228 " 6378 ! 6606
x2
y2
" 2 ! 1, we have c2 ! a2 # b2.
2
a
b
When x ! c,
y
2a ! 13,931
%
c2
y2
a2 # b2
" 2 ! 1 ⇒ y 2 ! b2 1 #
2
a
b
a2
a−c
a ! 6965.5
b
c ! 7325 # 6965.5
© Houghton Mifflin Company. All rights reserved.
! 359.5
e!
−a
c
c
' 0.0516
a
a
x
−b
⇒ y2 !
b4
a2
⇒ 2y !
2b2
.
a
&
a+c
56.
x2 y2
" !1
4
1
57.
y
x2
y2
"
!1
9
16
y
2
a ! 2, b ! 1, c ! !3
Points on the ellipse:
"± 2, 0#, "0, ± 1#
Length of latus recta:
(−
3, 12
)
(
−1
(−
3, − 12
(
−2
3, − 12
)
Points on the ellipse:
"± 3, 0#, "0, ± 4#
(− 94 , 7 )
−4
(− 49 , − 7 )
Length of latus recta:
( 49 , 7 )
2
−2
2
−2
2b2 2"3#2 9
!
!
a
4
2
2b2
!1
a
Additional points:
a ! 4, b ! 3, c ! !7
)
x
1
)
3, 12
%!3, ± 2&, %# !3, ± 2&
1
1
Additional points:
%± 4, # !7&, %± 4, !7&
9
9
4
x
( 49 , − 7 )
794
Chapter 9
Topics in Analytic Geometry
58. 9x2 " 4y2 ! 36
59. 5x2 " 3y 2 ! 15
y
(− 43 ,
x2 y2
" !1
4
9
Points on the ellipse:
"± 2, 0#, "0, ± 3#
5
)
−3
(
− 43 , −
Length of latus recta:
( 43 , 5 )
2
−1
)
1
−2
5
x2 y 2
"
!1
3
5
x
3
(
4
,−
3
5
−4
)
)
2
(3 5 5 , 2 )
−2
2
(− 3 5 5 , − 2 )
c ! !2
Length of latus recta:
4
Additional points:
%±
x
4
(3 5 5 , − 2 )
−4
Points on the ellipse:
"± !3, 0#, "0, ± !5#
%± 3, # !5&, %± 3, !5&
4
(− 3 5 5 ,
a ! !5, b ! !3,
2b2 2 % 22 8
!
!
a
3
3
Additional points:
y
4
2b2 2 % 3 6!5
!
!
!5
a
5
&%
3!5
3!5
, # !2 , ±
, !2
5
5
60. Answers will vary.
61. True. If e ' 1 then the ellipse is elongated, not
circular.
62. True. The ellipse is inside the circle.
63. (a) The length of the string is 2a.
&
(b) The path is an ellipse because the sum of the
distances from the two thumbtacks is always
the length of the string, that is, it is constant.
64. (a) a " b ! 20 ⇒ b ! 20 # a
(c)
A ! $ab ! $a"20 # a#
264 ! $a"20 # a#
(b)
$a2
# 20$a " 264 ! 0
(d)
a
8
9
10
11
12
13
A
301.6
311.0
314.2
311.0
301.6
285.9
360
a ' 14 or a ' 6 by the Quadratic Formula
Since a > b, we choose a ! 14 and b ! 6.
x2
y2
" 2!1
2
14
6
0
24
0
The area is maximum when a ! b ! 10 and
it is a circle.
x2
y2
"
!1
196 36
65. Center: "6, 2#
Foci: "2, 2#, "10, 2# ⇒ c ! 4
"a " c# " "a # c# ! 2a ! 36 ⇒ a ! 18
b2 ! a2 # b2 ⇒ b ! !182 # 16 ! !308
Horizontal major axis
"x # 6#2 " y # 2#2
"
!1
324
308
66.
y2
x2
" 2!1
2
a
b
The sum of the distances
from any point on the
ellipse to the two foci is
constant. Using the vertex
"a, 0#, you have
"a " c# " "a # c# ! 2a.
y
b2 + c2
b
b
c
−a
−c
c
−b
From the figure,
2!b2 " c2 ! 2a ⇒ a2 ! b2 " c2.
a
x
© Houghton Mifflin Company. All rights reserved.
b ! 64 ORb ! 14
Section 9.3
67. Arithmetic: d ! "11
71.
6
!3
n
72.
! 1093
n!0
Section 9.3
■
■
1
68. Geometric: r ! 2
6
! ""3#
n
69. Geometric: r ! 2
73.
! 547
10
! 4" #
n!1
n!0
3 n"1
$
4
15.099
Hyperbolas
70. Arithmetic: d ! 1
74.
10
! 5" #
n!0
■
© Houghton Mifflin Company. All rights reserved.
■
4 n
3
$ 340.155
Hyperbolas
A hyperbola is the set of all points "x, y# the difference of whose distances from two distinct fixed points
(foci) is constant.
The standard equation of a hyperbola with center "h, k# and transverse and conjugate axes of lengths
2a and 2b is:
"x " h#2 " y " k#2
"
! 1 if the transverse axis is horizontal.
(a)
a2
b2
" y " k#2 "x " h#2
"
! 1 if the transverse axis is vertical.
a2
b2
c2 ! a2 $ b2 where c is the distance from the center to a focus.
The asymptotes of a hyperbola are:
b
(a) y ! k ± "x " h# if the transverse axis is horizontal.
a
a
(b) y ! k ± "x " h# the transverse axis is vertical.
b
c
The eccentricity of a hyperbola is e ! .
a
To classify a nondegenerate conic from its general equation Ax2 $ Cy2 $ Dx $ Ey $ F ! 0:
(a) If A ! C (A # 0, C # 0), then it is a circle.
(b) If AC ! 0 (A ! 0 or C ! 0, but not both), then it is a parabola.
(c) If AC > 0, then it is an ellipse.
(d) If AC < 0, then it is a hyperbola.
(b)
■
■
Vocabulary Check
1. hyperbola
2. branches
4. asymptotes
5. Ax $ Cy $ Dx $ Ey $ F ! 0
1. Center: "0, 0#
2
3. transverse axis, center
2
2. Center: "0, 0#
a ! 3, b ! 5, c ! %34
a ! 5, b ! 3
Vertical transverse axis
Vertical transverse axis
Matches graph (b).
Matches graph (c).
3. Center: "1, 0#
4. Center: ""1, 2#
a ! 4, b ! 2
a ! 4, b ! 3
Horizontal transverse axis
Horizontal transverse axis
Matches graph (a).
Matches graph (d).
795
796
Chapter 9
Topics in Analytic Geometry
5. x2 " y2 ! 1
a ! 1, b ! 1, c ! %2
2
Center: "0, 0#
1
Vertices: "± 1, 0#
6.
y
10
8
6
4
a ! 3, b ! 5,
x
2
Asymptotes: y ! ± x
y
Center: "0, 0#
–2
Foci: "± %2, 0#
x2
y2
"
!1
9
25
–1
c ! %32 $ 52 ! %34
–2
Vertices: "± 3, 0#
x
−8 −6 −4
2 4 6 8 10
−6
−8
−10
Foci: "± %34, 0#
b
5
Asymptotes: y ! ± x ! ± x
a
3
y2 x2
" !1
1
4
8.
y
3
a ! 1, b ! 2, c ! %5
Center: "0, 0#
Vertices: "0, ± 1#
c ! %32 $ 12 ! %10
x
–2
2
Foci: "0, ± %5 #
3
Center: "0, 0#
–2
10.
x2
y2
" !1
36
4
a ! 5, b ! 9, c ! %a2 $ b2 ! %106
a ! 6, b ! 2,
Center: "0, 0#
c ! %36 $ 4 ! 2%10
y
15
12
9
Foci: "0, ± %106 #
−9 −6
"x " 1#2 " y $ 2#2
"
!1
4
1
y
6 9 12 15
Foci: "± 2%10, 0#
x
2
4
6
12.
1
x
1
1
Asymptotes: y ! "2 ± "x " 1#
2
8
4
x
–12
12
–4
–8
–12
"x $ 3#2 " y " 2#2
"
!1
144
25
2
3
a ! 12, b ! 5, c ! 13
Vertices: ""15, 2#, "9, 2#
–5
12
y
15
Center: ""3, 2#
2
–4
y
1
Asymptotes: y ! ± x
3
3
Center: "1, "2#
Foci: "1 ± %5, "2#
Vertices: "± 6, 0#
−3
−9
−12
−15
Vertices:
""1, "2#, "3, "2#
Center: "0, 0#
3
a
5
y!± x!± x
b
9
a ! 2, b ! 1, c ! %5
–2
Asymptotes: y ! ± 3x
Vertices: "0, ± 5#
11.
–4
–6
Foci: "0, ± %10#
y2
x2
"
!1
25 81
Asymptotes:
x
–6
Vertices: "0, ± 3#
–3
1
Asymptotes: y ! ± x
2
9.
y
6
a ! 3, b ! 1,
2
–3
y2 x2
" !1
9
1
Foci: ""16, 2#, "10, 2#
Asymptotes:
y!2 ±
5
"x $ 3#
12
10
5
−5
x
−5
−10
−15
−20
5
© Houghton Mifflin Company. All rights reserved.
7.
Section 9.3
13.
" y $ 5#2 "x " 1#2
"
!1
1(9
1(4
14.
%19 $ 14 !
1
1
a! ,b! ,c!
3
2
%13
6
Center: "1, "5#
Vertices:
Foci:
&1, "5 ± 13': &1, " 163', &1, " 143'
&1, "5
±
%13
6
Asymptotes: y ! k ±
Center: ""3, 1#
1
1
a! ,b! ,c!
2
4
Vertices:
−1
%14 $ 161 !
%5
4
&"3, 12', &"3, 32'
Foci:
a
"x " h#
b
Asymptotes: y ! 1 ±
&"3, 1 ± 45'
%
2
"x " 1#
3
1(2
"x $ 3# ! 1 ± 2"x $ 3#
1(4
y
3
y
−2
2
1
2
3
4
x
1
−1
x
−2
–3
–1
−3
–1
−5
15. (a) 4x 2 " 9y 2 ! 36
16. (a) 25x 2 " 4y 2 ! 100
x2 y 2
" !1
9
4
x2
y2
"
!1
4
25
© Houghton Mifflin Company. All rights reserved.
(b) Center: "0, 0#
(b) Center: "0, 0#
a ! 3, b ! 2, c ! %9 $ 4 ! %13
a ! 2, b ! 5, c ! %4 $ 25 ! %29
Vertices: "± 3, 0#
Vertices: "± 2, 0#
Foci: "± %13, 0#
Foci: "± %29, 0#
b
2
Asymptotes: y ! ± x ! ± x
a
3
b
5
Asymptotes: y ! ± x ! ± x
a
2
(c)
(c)
y
y
5
4
3
2
1
−5 −4
−2
8
6
4
2
−2
−3
−4
−5
4 5
797
" y " 1#2 "x $ 3#2
"
!1
1(4
1(16
'
y ! "5 ±
Hyperbolas
x
−8 −6 −4
4
−6
−8
6
8
x
798
Chapter 9
Topics in Analytic Geometry
17. (a) 2x2 " 3y2 ! 6
(c) To use a graphing calculator, solve first for y.
x2 y2
" !1
3
2
y2 !
(b) a ! %3, b ! %2, c ! %5
Vertices: "± %3, 0#
2
y2
Foci: "± %5, 0#
!±
%
2x " 6
! "%
3
2
!% x
3
2
! "% x
3
2x2 " 6
3
y1 !
Center: "0, 0#
Asymptotes: y ! ±
2x2 " 6
3
%23 x
y3
%6
y4
3
x
18. (a) 6y 2 " 3x 2 ! 18
(c)
)
)
y
4
Hyperbola
3
2
1
−4 −3
3
4
x
−2
Asymptotes
−3
−4
y
4
y2 x2
"
!1
3
6
3
1
(b) a ! %3, b ! %6, c ! 3
−4 −3 −2
Center: "0, 0#
2
−1
3
4
x
−3
Vertices: "0, ± %3#
−4
Foci: "0, ± 3#
19. (a)
%3
%6
x!±
%2
2
x
9x2 " y2 " 36x " 6y $ 18 ! 0
(c)
y
9"x2 " 4x $ 4# " " y2 $ 6y $ 9# ! "18 $ 36 " 9
"x " 2#
" y $ 3#
"
!1
1
9
2
2
x
2
–6 –4 –2
2
4
6
8
–4
–6
(b) a ! 1, b ! 3, c ! %10
–8
Center: "2, "3#
Vertices: "1, "3#, "3, "3#
Foci: "2 ± %10, "3#
Asymptotes: y ! "3 ± 3"x " 2#
20. (a) x2 " 9y2 $ 36y " 72 ! 0
x2 " 9" y2 " 4y $ 4# ! 72 " 36
(b) a ! 6, b ! 2,
(c)
y
12
c ! %36 $ 4 ! 2%10
x2 " 9" y " 2#2 ! 36
Center: "0, 2#
x2
" y " 2#2
"
!1
36
4
Vertices: "± 6, 2#
Foci: "± 2%10, 2#
1
Asymptotes: y ! 2 ± x
3
8
4
x
–8
–4
–4
–8
–12
4
8
© Houghton Mifflin Company. All rights reserved.
Asymptotes: y ! ±
Section 9.3
21. (a)
(c)
x2 " 9y2 $ 2x " 54y " 80 ! 0
y
4
"x2 $ 2x $ 1# " 9" y2 $ 6y $ 9# ! 80 $ 1 " 81
2
"x $ 1#2 " 9" y $ 3#2 ! 0
y$3!
x
1
± 3 "x
–4
$ 1#
–2
–2
–4
(b) Degenerate hyperbola is two lines intersecting at
""1, "3#.
22. (a)
16" y2 $ 4y $ 4# " "x2 " 2x $ 1# ! "63 $ 64 " 1
y
x
–1
1
2
3
–1
16" y $ 2#2 " "x " 1# ! 0
y $ 2 ! ± 14"x " 1#
(b) Degenerate hyperbola is two intersecting lines at
"1, "2#.
23. (a)
–6
(c)
16y2 " x2 $ 2x $ 64y $ 63 ! 0
2
–2
–3
–4
9y2 " x2 $ 2x $ 54y $ 62 ! 0
9" y2 $ 6y $ 9# " "x2 " 2x $ 1# ! "62 " 1 $ 81
" y $ 3#2 "x " 1#2
"
!1
2
18
(b) a ! %2, b ! 3%2, c ! 2%5
Center: "1, "3#
Vertices: "1, "3 ± %2 #
Foci: "1, "3 ± 2%5 #
Asymptotes: y ! "3 ±
1
"x " 1#
3
(c) To use a graphing calculator, solve for y first.
4
© Houghton Mifflin Company. All rights reserved.
9" y $ 3#2 ! 18 $ "x " 1#2
y ! "3 ±
%
y
2
18 $ "x " 1#2
9
1
y1 ! "3 $ %18 $ "x " 1#2
3
1
y2 ! "3 " %18 $ "x " 1#2
3
1
y3 ! "3 $ "x " 1#
3
1
y4 ! "3 " "x " 1#
3
2
)
)
−6
−8
Hyperbola
Asymptotes
−10
x
Hyperbolas
799
Chapter 9
800
24. (a)
Topics in Analytic Geometry
(c)
9x2 " y2 $ 54x $ 10y $ 55 ! 0
y
14
9"x2 $ 6x $ 9# " " y 2 " 10y $ 25 # ! "55 $ 81 " 25
10
"x $ 3#2 " y " 5#2
"
!1
1(9
1
8
6
4
%10
1
(b) a ! , b ! 1, c !
3
3
2
−7 −6
−4 −3 −2
1
x
Center: ""3, 5#
&"3 ± 3, 5'
1
Vertices:
Foci:
&"3 ±
%10
3
,5
'
Asymptotes: y ! 5 ± 3"x $ 3#
25. Vertices: "0, ± 2# ⇒ a ! 2
26. Vertices: "± 3, 0# ⇒ a ! 3
Foci: "0, ± 4# ⇒ c ! 4
Foci: "± 6, 0# ⇒ c ! 6
2
2
2
2
2
Asymptotes:
b ! c " a ! 16 " 4 ! 12
b ! c " a ! 36 " 9 ! 27
Center: "0, 0# ! "h, k#
x2
y2
" 2!1
2
a
b
" y " k#2 "x " h#2
"
!1
a2
b2
x2
y2
"
!1
9
27
y2
x2
"
!1
4
12
28. Vertices: "0, ± 3# ⇒ a ! 3
Asymptotes: y ! ± 3x ⇒
a
! 3, b ! 1
b
y ! ± 5x ⇒
⇒ b!5
Center: "0, 0#
x2
y2
"
!1
1
25
29. Foci: "0, ± 8# ⇒ c ! 8
Asymptotes: y ! ± 4x ⇒
a
! 4 ⇒ a ! 4b
b
Center: "0, 0# ! "h, k#
Center: "0, 0# ! "h, k#
" y " k#
"x " h#
"
!1
2
a
b2
c2 ! a2 $ b2 ⇒ 64 ! 16b2 $ b2
2
2
y2
" x2 ! 1
9
b
!5
a
64
1024
! b2 ⇒ a2 !
17
17
" y " k#2 "x " h#2
"
!1
a2
b2
y2
x2
"
!1
1024(17 64(17
17y2
17x2
"
!1
1024
64
© Houghton Mifflin Company. All rights reserved.
2
27. Vertices: "± 1, 0# ⇒ a ! 1
Section 9.3
31. Vertices: "2, 0#, "6, 0# ⇒ a ! 2
30. Foci: "± 10, 0# ⇒ c ! 10
Foci: "0, 0#, "8, 0# ⇒ c ! 4
3
b 3m
Asymptotes: y ! ± x ⇒ !
4
a 4m
c2
!
a2
$
b2
Hyperbolas
b2 ! c2 " a2 ! 16 " 4 ! 12
⇒ 100 ! "3m# $ "4m#
2
2
100 ! 25m
2!m
a ! 4"2# ! 8, b ! 3"2# ! 6
x2
y2
" 2!1
2
a
b
2
Center: "4, 0# ! "h, k#
"x " h#2 " y " k#2
"
!1
a2
b2
"x " 4#2 y 2
"
!1
4
12
y2
x2
"
!1
64 36
32. Vertices: "2, 3#, "2, "3# ⇒ a ! 3
Center: "2, 0#
Foci: "4, 0#, "4, 10# ⇒ c ! 5
Foci: "2, 5#, "2, "5# ⇒ c ! 5
b2 ! c2 " a2 ! 25 " 16 ! 9
b2 ! c2 " a2 ! 25 " 9 ! 16
Center: "4, 5# ! "h, k#
" y " k#2 "x " h#2
"
!1
a2
b2
" y " k#2 "x " h#2
"
!1
a2
b2
y2 "x " 2#2
"
!1
9
16
" y " 5#2 "x " 4#2
"
!1
16
9
34. Vertices: ""2, 1#, "2, 1# ⇒ a ! 2
© Houghton Mifflin Company. All rights reserved.
33. Vertices: "4, 1#, "4, 9# ⇒ a ! 4
35. Vertices: "2, 3#, "2, "3# ⇒ a ! 3
Center: "0, 1#
Solution point: "0, 5#
Foci: ""3, 1#, "3, 1# ⇒ c ! 3
Center: "2, 0# ! "h, k#
b2 ! c2 " a2 ! 9 " 4 ! 5
" y " k#2 "x " h#2
"
!1
a2
b2
"x " h#2 " y " k#2
"
!1
a2
b2
x2 " y " 1#2
"
!1
4
5
y2 "x " 2#2
"
!1 ⇒
9
b2
b2 !
!
9"x " 2#2
y2 " 9
9""2#2 36 9
!
!
25 " 9 16 4
y2 "x " 2#2
"
!1
9
9(4
801
Topics in Analytic Geometry
37. Vertices: "0, 4#, "0, 0#
36. Center: "0, 1#, a ! 2
Center: "0, 2#, a ! 2
x 2 " y " 1#2
"
!1
4
b2
" y " 2#2 x 2
" 2!1
4
b
Solution point: "5, 4#
Passes through "%5, "1#
25
9
" 2!1
4
b
""1 " 2#2
5
" 2!1
4
b
9
21
!
2
b
4
9
5
"1! 2
4
b
36 12
b2 !
!
21
7
x2
4
b2 ! 4 ⇒ b ! 2
" y " 1#
!1
12(7
2
"
38. Center: "1, 0#, a ! 2
y 2 "x " 1#2
"
!1
4
b2
Solution point: "0, %5#
5
1
" !1
4 b2
1
1
! ⇒ b!2
b2 4
y 2 "x " 1#2
"
!1
4
4
" y " 2#2 x2
" !1
4
4
39. Vertices:
40. Center: "3, "3#, a ! 3
"1, 2#, "3, 2# ⇒ a ! 1
Asymptotes:
Center: "2, 2#
y ! x " 6, y ! "x
Asymptotes:
1!
y ! x, y ! 4 " x
a 3
! ⇒ b!3
b b
" y $ 3#2 "x " 3#2
"
!1
9
9
b
!1 ⇒ b!1
a
"x " 2#2 " y " 2#2
"
!1
1
1
41. Vertices: "0, 2#, "6, 2# ⇒ a ! 3
42. Vertices: (3, 0#, "3, 4# ⇒ a ! 2
2
2
Asymptotes: y ! x, y ! 4 " x
3
3
2
2
Asymptotes: y ! x, y ! 4 " x
3
3
b 2
! ⇒ b!2
a 3
a 2
! ⇒ b!3
b 3
Center: "3, 2# ! "h, k#
Center: "3, 2# ! "h, k#
"x " h#2 " y " k#2
"
!1
a2
b2
" y " k#2 "x " h#2
"
!1
a2
b2
"x " 3#2 " y " 2#2
"
!1
9
4
" y " 2#2 "x " 3#2
"
!1
4
9
© Houghton Mifflin Company. All rights reserved.
Chapter 9
802
Section 9.3
43. F1: Friend’s location ""10,560, 0#
Hyperbolas
y
F2: Your location "10,560, 0#
10,000
P
P"x, y#: Location of lightning strike
Friend F1
"1100#"18# ! 19,800
F2
−20,000
x2
y2
2 " 2 ! 1
a
b
You
20,000
x
(−10,560, 0) (10,560, 0)
−10,000
19,800
! 9900 ⇒ a2 ! 98,010,000
2
c ! 10,560, a !
b2 ! c2 " a2 ! 13,503,600
y2
x2
"
98,010,000 13,503,600
44. The explosion occurred on the vertical line through
"3300, 1100# and "3300, 0#.
y
4000
(3300, 1100)
3000
d2 " d1 ! 4"1100# ! 4400
Hence,
(3300, 0)
2000
(−3300, 0)
1000
x
−4000
a
2a ! 4400
d2
a ! 2200
d1
−4000
c ! 3300
b2 ! c2 " a2.
The explosion occurred on the hyperbola
x2
y2
" 2 ! 1.
2
a
b
Letting x ! 3300,
y 2 ! b2
&ax
2
2
'
" 1 ! "33002 " 22002#
&3300
2200
2
2
"1
' ⇒ y ! "2750.
© Houghton Mifflin Company. All rights reserved.
"3300, "2750#
45. (a)
x2
y2
"
!1
a2 b2
a ! 1; "2, 9# is on the curve, so
4 81
81
" 2 !1 ⇒ 2 !3
1
b
b
81
⇒ b2 !
⇒ b ! 3%3.
3
x2
1
"
y2
27
(b) Because each unit is 21 foot, 4 inches is 23 of a unit.
The base is 9 units from the origin, so
y!9"
2
1
!8 .
3
3
When y !
x2 ! 1 $
! 1, "9 ≤ y ≤ 9
25
,
3
"25(3#2
⇒ x $ 1.88998.
27
So the width is 2x $ 3.779956 units, or
22.68 inches, or 1.88998 feet.
803
804
Chapter 9
Topics in Analytic Geometry
46. Foci: "± 150, 0# ⇒ c ! 150
Center: "0, 0#
y
150
(x, 75)
75
d2
d1
(−150, 0)
−150
(150, 0)
x
−75
75
150
(a) d2 " d1 ! "186,000#"0.001#
(c) Bay to Station 1: 30 miles
! 186 ⇒ 2a ! 186 ⇒ a ! 93
b2 ! c2 " a2 ! 1502 " 932 ! 13,851
x2
y2
"
!1
2
93
13,851
Bay to Station 2: 270 miles
"270 " 30#
$ 0.00129 second
186,000
(d) In this case,
&
x2 ! 932 1 $
'
752
$ 12,161.43
13,851
x $ 110.3 miles
(b) 150 " 93 ! 57 miles
d2 " d1 ! 186,000"0.00129# $ 239.94 ⇒ a $ 120
and b2 ! c2 " a2 ! 8100. The hyperbola is
x2
y2
"
! 1.
1202 902
For y ! 60, x 2 ! 20,800 and x $ 144.2.
Position: "144.2, 60#
47. Center: "0, 0#
Focus: "24, 0#
b2 ! c2 " a2 ! 242 " a2 ! 576 " a2
242
242
"
!1
a2
576 " a2
576
576
"
!1
a2
576 " a2
576"576 " a2# " 576a2 ! a2"576 " a2#
a4 " 1728a2 $ 331,776 ! 0
a $ ± 38.83 or a $ ± 14.83
Since a < c and c ! 24, we choose a ! 14.83. The vertex is approximate at "14.83, 0#.
[Note: By the Quadratic Formula, the exact value of a is a ! 12"%5 " 1#.]
48.
x2
y2
"
!1
25 16
a ! 5, b ! 4, c ! %25 $ 16 ! %41
The camera is 5 $ %41 units from the mirror.
49. 9x 2 $ 4y 2 " 18x $ 16y " 119 ! 0
A ! 9, C ! 4
AC ! 36 > 0, Ellipse
© Houghton Mifflin Company. All rights reserved.
x2
y2
"
!1
a2 576 " a2
Section 9.3
50. x 2 $ y 2 " 4x " 6y " 23 ! 0
Hyperbolas
51. 16x 2 " 9y 2 $ 32x $ 54y " 209 ! 0
A ! C ! 1, Circle
A ! 16, C ! "9
AC ! 16""9# < 0, Hyperbola
52. x 2 $ 4x " 8y $ 20 ! 0
53. y 2 $ 12x $ 4y $ 28 ! 0
A ! 1, C ! 0
C ! 1, A ! 0
AC ! 0, Parabola
AC ! 0, Parabola
54. 4x 2 $ 25y 2 $ 16x $ 250y $ 541 ! 0
55. x2 $ y 2 $ 2x " 6y ! 0
A ! 4, C ! 25
A ! C ! 1, Circle
AC ! 100 > 0, Ellipse
56. y 2 " x 2 $ 2x " 6y " 8 ! 0
57. x 2 " 6x " 2y $ 7 ! 0
A ! "1, C ! 1
A ! 1, C ! 0, D ! "6, E ! "2, F ! 7
AC < 0, Hyperbola
AC ! 0 ⇒ Parabola
59. True. e !
58. 9x 2 $ 4y 2 " 90x $ 8y $ 228 ! 0
%a2 $ b2
c
!
a
a
A ! 9, C ! 4
AC ! 9"4# ! 36 > 0 ⇒ Ellipse
60. False. b # 0 because it is in the denominator.
61. False. For example,
2
62. True. The asymptotes are
2
x " y " 2x $ 2y ! 0
b
y ! ± x.
a
"x " 1#2 " " y " 1#2 ! 0
If they intersect at right angles, then
© Houghton Mifflin Company. All rights reserved.
is the graph of two intersecting lines.
b
a
"1
! ⇒ a ! b.
!
a ""b(a# b
63. Let "x, y# be such that the difference of the distances from "c, 0# and ""c, 0#
is 2a (again only deriving one of the forms).
*
*
2a ! %"x $ c#2 $ y2 " %"x " c# $ y2
2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2
4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2
4a%"x " c#2 $ y2 ! 4cx " 4a2
a%"x " c#2 $ y2 ! cx " a2
a2"x2 " 2cx $ c2 $ y2# ! c2x2 " 2a2cx $ a4
a2"c2 " a2# ! "c2 " a2#x2 " a2y2
Let b2 ! c2 " a2. Then a2b2 ! b2x2 " a2y2 ⇒ 1 !
x2
y2
" 2.
2
a
b
805
806
Chapter 9
Topics in Analytic Geometry
*
64. Answers will vary. See Example 3.
*
65. d2 " d1 ! constant by definition of hyperbola
At the point "a, 0#,
*d2 " d1* ! *"a $ c# " "c " a#* ! 2a.
66. Center: "6, 2#
Horizontal transverse axis
Foci at "2, 2# and "10, 2# ⇒ c ! 4.
"c $ a# " "c " a# ! 6 ⇒ a ! 3
b2 ! c2 " a2 ! 16 " 9 ! 7
"x " 6#2 " y " 2#2
"
!1
9
7
67. At the point "a, 0#, the difference of the distances to the foci "± c, 0# is "c $ a# " "c " a# ! 2a.
Let "x, y# be a point on the hyperbola.
2a ! %"x $ c#2 $ y2 " %"x " c#2 $ y2
2a $ %"x " c#2 $ y2 ! %"x $ c#2 $ y2
4a2 $ 4a%"x " c#2 $ y2 $ "x " c#2 $ y2 ! "x $ c#2 $ y2
4a%"x " c#2 $ y2 ! 4cx " 4a2
a%"x " c#2 $ y2 ! cx " a2
a2"x 2 " 2cx $ c2 $ y2# ! c2x 2 " 2a2cx $ a4
a2"c2 " a2# ! "c2 " a2#x 2 " a2y 2
1!
x2
y2
" 2
2
a
c " a2
Thus, c2 " a2 ! b2, as desired.
If A ! 0 and C # 0, then Cy 2 $ Dx $ Ey $ F ! 0 is a parabola (complete the square).
Same for A # 0 and C ! 0.
If AC > 0, then both A and C are positive (or both negative). By completing the square
you obtain an ellipse.
If AC < 0, then A and C have opposite signs. You obtain a hyperbola.
69. "x3 " 3x 2# " "6 " 2x " 4x 2# ! x3 $ x 2 $ 2x " 6
1
1
70. "3x " 2 #"x $ 4# ! 3x2 $ 12x " 2x " 2
! 3x2 $
71. "2
1
0
"2
"3
4
4
"2
1 "2
1
2
x3 " 3x $ 4
2
! x 2 " 2x $ 1 $
x$2
x$2
23
2x
"2
2
2
72. +"x $ y# $ 3, ! "x $ y# $ 6"x $ y# $ 9
! x2 $ 2xy $ y2 $ 6x $ 6y $ 9
© Houghton Mifflin Company. All rights reserved.
68. If A ! C # 0, then by completing the square you obtain a circle.
Section 9.4
Rotation and Systems of Quadratic Equations
73. x3 ! 16x " x!x 2 ! 16" " x!x ! 4"!x # 4"
74. x2 # 14x # 49 " !x # 7"2
75. 2x3 ! 24x 2 # 72x " 2x!x 2 ! 12x # 36"
76. 6x3 ! 11x 2 ! 10x " x!6x2 ! 11x ! 10"
" 2x!x ! 6"2
" x!3x # 2"!2x ! 5"
77. 16x3 # 54 " 2!8x3 # 27"
78. 4 ! x # 4x2 ! x3 " !4 ! x" # x2!4 ! x"
" 2!2x # 3"!4x 2 ! 6x # 9"
" !4 ! x"!x2 # 1"
" !4 ! x"!x # i"!x ! i"
Section 9.4
■
■
■
Rotation and Systems of Quadratic Equations
Ax2 # Bxy # Cy2 # Dx # Ey # F " 0
The general second-degree equation
can be rewritten
2
2
A$ !x$ " # C$! y$ " # D$x$ # E$y$ # F$ " 0 by rotating the coordinate axes through the angle %
where cot 2% " !A ! C "#B.
x " x$ cos % ! y$sin %
y " x$ sin % # y$ cos %
The graph of the nondegenerate equation Ax2 # Bxy # Cy2 # Dx # Ey # F " 0 is:
(a) An ellipse or circle if B2 ! 4AC < 0.
(b) A parabola if B2 ! 4AC " 0.
(c) A hyperbola if B2 ! 4AC > 0.
Vocabulary Check
1. rotation, axes
2. invariant under rotation
© Houghton Mifflin Company. All rights reserved.
1. % " 90&; Point: !0, 3"
x " x$ cos % ! y$ sin %
y " x$ sin % # y$ cos %
0 " x$ cos 90& ! y$ sin 90&
3 " x$ sin 90& # y$ cos 90&
0 " y$
3 " x$
Thus, !x$, y$ " " !3, 0".
2. % " 45&; Point: !3, 3"
x " x$ cos % ! y$ sin %
y " x$ sin % # y$ cos %
3 " x$ cos 45& ! y$ sin 45&
3 " x$ sin 45& # y$ cos 45&
3"
$2
2
x$ !
$2
2
y$
Adding, 6 " $2x$ ⇒ x$ "
3"
2
x$ #
6
" 3$2.
$2
Subtracting, $2y$ " 0 ⇒ y$ " 0.
Thus, !x$, y$ " " !3$2, 0".
$2
$2
2
y$
3. discriminant
807
Chapter 9
808
Topics in Analytic Geometry
3. xy # 1 " 0
y
y′
A " 0, B " 1, C " 0
cot 2% "
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
B
2
4
$2
$2
'
'
x$ ! y$
! y$ sin " x$
! y$
"
4
4
2
2
$2
$
x′
−4 −3 −2
% & % &
'
'
2
2
x$ # y$
y " x$ sin # y$ cos " x$ %
# y$ %
"
4
4
2 &
2 &
2
x " x$ cos
4
4
x
−2
−3
−4
$
$
xy # 1 " 0
%
&%
x$ ! y$
$2
x$ # y$
$2
& #1"0
! y$ "2 !x$ "2
!
" 1, Hyperbola
2
2
4. xy ! 2 " 0, A " 0, B " 1, C " 0
cot 2% "
y
y′
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
B
2
4
% & % &
'
'
2
2
x$ # y$
y " x$ sin # y$ cos " x$ %
# y$ %
"
4
4
2 &
2 &
2
x " x$ cos
10
8
6
4
$2
$2
'
'
x$ ! y$
! y$ sin " x$
! y$
"
4
4
2
2
$2
$
$
$
x′
x
4 6 8 10
−8
−10
xy ! 2 " 0
%
x$ ! y$
$2
&%
&
x$ # y$
!2"0
$2
!x$ "2 ! y$ "2
!
" 1, Hyperbola
4
4
5. x 2 ! 4xy # y 2 # 1 " 0
y
8
y′
x′
6
4
−8 −6 −4
4
y " −6
x$ sin
−8
6
8
x
A " 1, B " !4, C " 1
'
'
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
# y$ coscot 2% "
B
2
4
4
4
x " x$ cos
" x$
"
% 22 & ! y$ % 22 &
$
$2
2
!x$ ! y$ "
$
'
'
! y$ sin
4
4
© Houghton Mifflin Company. All rights reserved.
!x$ "2 ! ! y$ "2
"2
2
Section 9.4
Rotation and Systems of Quadratic Equations
5. —CONTINUED—
x 2 ! 4xy # y 2 # 1 " 0
)
$2
2
* )
!x$ ! y$ "
2
!4
$2
2
!x$ ! y$ "
$2
2
* )
!x$ # y$ " #
$2
2
*
!x$ # y$ "
2
#1"0
1
1
1
1
!x$ "2 ! x$y$ # ! y$ "2 ! 2'!x$ "2 ! ! y$ "2( # !x$ "2 # x$y$ # ! y$ "2 # 1 " 0
2
2
2
2
! !x$ "2 # 3! y$ "2 " !1
!x$ "2 !
! y$ "2
" 1 , Hyperbola
1#3
6. xy # x ! 2y # 3 " 0
y
y′
A " 0, B " 1, C " 0
cot 2% "
" x$
"
6
'
'
! y$ sin
4
4
y " x$ sin
% 2 & ! y$ % 2 &
$2
$2
" x$
x$ ! y$
$2
x′
4
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
B
2
4
x " x$ cos
8
x
−8 −6 −4
'
'
# y$ cos
4
4
4
−4
−6
−8
% 2 & # y$ % 2 &
$2
$2
x$ # y$
$2
"
xy # x ! 2y # 3 " 0
%
x$ ! y$
$2
&%
& %
&
%
&
x$ # y$
x$ ! y$
x$ # y$
#
!2
#3"0
$2
$2
$2
!x$ "2 ! y$ "2
x$
y$
2x$
2y$
!
#
!
!
!
#3"0
2
2
$2
$2
$2
$2
)! $ "
© Houghton Mifflin Company. All rights reserved.
x
2
! $2x$ #
% 22 & * ! )! y$ " # 3 2y$ # %3 2 2 & * " !6 # % 22 & ! %3 2 2 &
%x$ ! 22 & ! %y$ # 3 2 2 & " !10
%y$ # 3 2 2 & ! %x$ ! 22 & " 1 , Hyperbola
$
2
2
$
$
$
10
2
2
$
2
$
$
2
$
2
10
2
$
2
6
809
810
Chapter 9
Topics in Analytic Geometry
7. xy ! 2y ! 4x " 0
y
8
A " 0, B " 1, C " 0
cot 2% "
" x$
"
y′
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
B
2
4
x " x$ cos
'
'
! y$ sin
4
4
y " x$ sin
% 22 & ! y$ % 22 &
$
x′
6
$
" x$
x$ ! y$
$2
"
'
'
# y$ cos
4
4
4
−4
2
4
6
8
x
−4
% 22 & # y$ % 22 &
$
$
x$ # y$
$2
xy ! 2y ! 4x " 0
%
x$ ! y$
$2
&%
x$ # y$
$2
& ! 2%
&
%
&
x$ # y$
x$ ! y$
!4
"0
$2
$2
!x$ "2 ! y$ "2
!
! $2x$ ! $2y$ ! 2$2x$ # 2$2y$ " 0
2
2
'!x$ "2 ! 6$2x$ # !3$2 "2( ! '! y$ "2 ! 2$2y$ # !$2 "2( " 0 # !3$2 "2 ! !$2 " 2
!x$ ! 3$2 "2 ! !y$ ! $2 "2 " 16
!x$ ! 3$2"2 ! ! y$ ! $2 "2 " 1, Hyperbola
16
16
8. 2x2 ! 3xy ! 2y2 # 10 " 0
y
A " 2, B " !3, C " !2
(!C
4
" ! ⇒ % + 71.57&
B
3
cos 2% " !
−4
4
5
2%
1 ! !!4#5"
"$
"
$1 ! cos
2
2
1 # cos 2%
1 # !!4#5"
cos % " $
"$
"
2
2
x " x$ cos % ! y$ sin %
"
% 110& ! y$ % 310&
$
x$ ! 3y$
$10
—CONTINUED—
x
−2
4
−4
sin % "
" x$
2
y′
$
3
$10
1
$10
y " x$ sin % # y$ cos %
" x$
"
% 310& # y$ % 110&
$
3x$ # y$
$10
$
© Houghton Mifflin Company. All rights reserved.
cot 2% "
x′
4
Section 9.4
Rotation and Systems of Quadratic Equations
8. —CONTINUED—
2x2 ! 3xy ! 2y2 # 10 " 0
2
%
&
x$ ! 3y$
$10
2
!3
%
x$ ! 3y$
$10
&%
&
%
3x$ # y$
3x$ # y$
!2
$10
$10
&
2
# 10 " 0
!x$ "2 6x$ y$ 9! y$ "2 9!x$ "2 24x$ y$ 9! y$ "2 9!x$ "2 6x$ y$ ! y$ "2
!
#
!
#
#
!
!
!
# 10 " 0
5
5
5
10
10
10
5
5
5
5
5
! !x$ "2 # ! y$ "2 " !10
2
2
!x$ "2 ! y$ "2
!
" 1 , Hyperbola
4
4
9. 5x2 ! 6xy # 5y2 ! 12 " 0
y
A " 5, B " !6, C " 5
cot 2% "
x′
3
2
A!C
'
'
" 0 ⇒ 2% "
⇒ %"
B
2
4
x " x$ cos
4
y′
x
−4 −3
2
'
' $2
! y$ sin "
!x$ ! y$ "
4
4
2
3
4
−3
−4
'
' $2
y " x$ sin # y$ cos "
!x$ # y$ "
4
4
2
5x2 ! 6xy # 5y2 ! 12 " 0
)
5
$2
2
* )
2
!x$ ! y$ "
!6
$2
2
!x$ ! y$ "
$2
2
* )
!x$ # y$ " # 5
$2
2
*
2
!x$ # y$ "
" 12
5
5
5
5
!x$ "2 ! 5x$y$ # ! y$ "2 ! 3!x$ "2 # 3! y$ "2 # !x$ "2 # 5x$y$ # ! y$ "2 " 12
2
2
2
2
2!x$ "2 # 8! y$ "2 " 12
© Houghton Mifflin Company. All rights reserved.
!x$ "2 ! y$ "2
#
" 1, Ellipse
6
3#2
10. 13x2 # 6$3xy # 7y2 ! 16 " 0
y
y′
3
A " 13, B " 6$3, C " 7
cot 2% "
A!C
1
'
'
"
⇒ 2% "
⇒ %"
B
$3
3
6
x " x$ cos
" x$
"
x′
'
'
! y$ sin
6
6
% 23 & ! y$ %12&
$
$3x$ ! y$
2
—CONTINUED—
y " x$ sin
" x$
"
−3
'
'
# y$ cos
6
6
%2& # y$ % 2 &
1
x$ # $3y$
2
$3
x
−2
2
−2
−3
3
811
Chapter 9
812
Topics in Analytic Geometry
10. —CONTINUED—
13x2 # 6$3xy # 7y2 ! 16 " 0
%
13
$3x$ ! y$
2
& # 6 3%
2
$
$3x$ ! y$
2
&%x$ # 2 3y$ & # 7%x$ # 2 3y$ & ! 16 " 0
$
2
$
39!x$ "2 13$3x$ y$ 13! y$ "2 18!x$ "2 18$3x$ y$ 6$3x$ y$
!
#
#
#
!
4
2
4
4
4
4
!
18! y$ "2 7!x$ "2 7$3x$ y$ 21! y$ "2
#
#
#
! 16 " 0
4
4
2
4
16!x$ "2 # 4! y$ "2 " 16
!x$ "2 ! y$ "2
#
" 1 , Ellipse
1
4
11. 3x2 ! 2$3xy # y2 # 2x # 2$3y " 0
y
2
A " 3, B " !2$3, C " 1
cot 2% "
A!C
1
"!
⇒ % " 60&
B
$3
−6
−4
2
%2& ! y$ % 2 & "
1
$3
y " x$ sin % # y$ cos % "
x
−2
x " x$ cos 60& ! y$ sin 60&
" x$
x′
y′
−4
x$ ! $3y$
2
$3x$ # y$
2
3x2 ! 2$3xy # y2 # 2x # 2$3y " 0
$
2
$
$
$3x$ # y$
2
&#%
& # 2%x$ ! 2 3y$ &
3x$ # y$
# 2 3%
&"0
2
$3x$ # y$
2
2
$
$
$
3(x$ ) 2 6$3x$y$ 9(y$ ) 2 6(x$ ) 2 4$3x$y$ 6(y$ ) 2 3(x$ ) 2 2$3x$y$ (y$ ) 2
!
#
!
#
#
#
#
#
4
4
4
4
4
4
4
4
4
# x$ ! $3y$ # 3x$ # $3y$ " 0
4(y$ ) 2 # 4x$ " 0
x$ " ! ! y$ " 2 ,
Parabola
© Houghton Mifflin Company. All rights reserved.
%x$ ! 2 3y$ & ! 2 3%x$ ! 2 3y$ &%
3
Section 9.4
Rotation and Systems of Quadratic Equations
12. 16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0
y
A " 16, B " !24, C " 9
cot 2% "
813
x′
(!C
7
"!
⇒ % + 53.13&
B
24
7
25
cos 2% " !
1
y′
1
2%
1 ! !!7#25" 4
"$
"
$1 ! cos
2
2
5
1 # cos 2%
1 # !!7#25" 3
cos % " $
"$
"
2
2
5
2
3
4
5
6
x
sin % "
x " x$ cos % ! y$ sin % " x$
%5& ! y$ %5& "
3x$ ! 4y$
5
y " x$ sin % # y$ cos % " x$
%5& # y$ %5& "
4x$ # 3y$
5
3
4
4
3
16x2 ! 24xy # 9y2 ! 60x ! 80y # 100 " 0
16
%
3x$ ! 4y$
5
& ! 24%
2
3x$ ! 4y$
5
&%
& %
4x$ # 3y$
4x$ # 3y$
#9
5
5
& ! 60%
2
&
%
&
3x$ ! 4y$
4x$ # 3y$
! 80
# 100 " 0
5
5
144!x$ "2 384x$ y$
256! y$ "2 288!x$ "2 168x$ y$
288! y$ "2 144!x$ "2 216x$ y$
!
#
!
#
#
#
#
25
25
25
25
25
25
25
25
#
81! y$ "2
! 36x$ # 48y$ ! 64x$ ! 48y$ # 100 " 0
25
25! y$ "2 ! 100x$ # 100 " 0
! y $ "2 " 4!x$ ! 1"
Parabola
13. 9x2 # 24xy # 16y2 # 90x ! 130y " 0
y
6
© Houghton Mifflin Company. All rights reserved.
A " 9, B " 24, C " 16
A!C
7
cot 2% "
"!
⇒ % + 53.13&
B
24
cos 2% " !
y′
2
7
25
−4
$
$
1 # cos 2%
1 # !!7#25" 3
cos % " $
"$
"
2
2
5
sin % "
x " x$ cos % ! y$ sin %
" x$
"
%5& ! y$ %5&
3
2
−2
1 ! cos 2%
"
2
4
3x$ ! 4y$
5
—CONTINUED—
1 ! !!7#25" 4
"
2
5
y " x$ sin % # y$ cos %
" x$
"
%5& # y$ %5&
4
4x$ # 3y$
5
x′
4
3
4
x
Chapter 9
814
Topics in Analytic Geometry
13. —CONTINUED—
9x2 # 24xy # 16y2 # 90x ! 130y " 0
9
%
3x$ ! 4y$
5
&
2
# 24
%
3x$ ! 4y$
5
&%
&
%
4x$ # 3y$
4x$ # 3y$
# 16
5
5
&
2
# 90
! 130
%
3x$ ! 4y$
5
&
%4x$ #5 3y$ & " 0
81!x$ "2 216x$y$ 144! y$ "2 288!x$ "2 168x$y$ 288! y$ "2 256!x$ "2 384x$y$
!
#
#
!
!
#
#
25
25
25
25
25
25
25
25
#
144! y$"2
# 54x$ ! 72y$ ! 104x$ ! 78y$ " 0
25
25!x$ "2 ! 50x$ ! 150y$ " 0
!x$ "2 ! 2x$ # 1 " 6y$ # 1
%
!x$ ! 1"2 " 6 y$ #
14. 9x2 # 24xy # 16y2 # 80x ! 60y " 0
&
1
, Parabola
6
y
A " 9, B " 24, C " 16
3
y′
cot 2% "
(!C
7
"!
⇒ % + 53.13&
B
24
cos 2% " !
2
x′
1
7
25
−3
−2
x
−1
1
−1
2%
1 ! !!7#25" 4
"$
"
$1 ! cos
2
2
5
1 # cos 2%
1 # !!7#25" 3
cos % " $
"$
"
2
2
5
sin % "
y " x$ sin % # y$ cos %
x " x$ cos % ! y$ sin % " x$
%5& ! y$ %5& "
3x$ ! 4y$
5
y " x$ sin % # y$ cos % " x$
%5& # y$ %5& "
4x$ # 3y$
5
3
4
4
3
9x2 # 24xy # 16y2 # 80x ! 60y " 0
9
%
3x$ ! 4y$
5
&
2
# 24
%
3x$ ! 4y$
5
&%
&
%
4x$ # 3y$
4x$ # 3y$
# 16
5
5
&
2
# 80
%
&
%
&
3x$ ! 4y$
4x$ # 3y$
! 60
"0
5
5
81!x$ "2 216x$ y$
144! y$ "2 288!x$ "2 168x$ y$
288! y$ "2 256!x$ "2 384x$ y$
!
#
#
!
!
#
#
25
25
25
25
25
25
25
25
#
144! y$ "2
# 48x$ ! 64y$ ! 48x$ ! 36y$ " 0
25
25!x$ "2 ! 100y$ " 0
!x $ "2 " 4y$ ,
Parabola
© Houghton Mifflin Company. All rights reserved.
x " x$ cos % ! y$ sin %
Section 9.4
Rotation and Systems of Quadratic Equations
815
15. x 2 # 3xy # y 2 " 20
cot 2% "
A!C 1!1
'
"
" 0 ⇒ % " " 45&
B
3
4
Solve for y in terms of x:
Graph y1 " !
y 2 # 3xy " 20 ! x 2
y 2 # 3xy #
%
8
9x2
9x2
" 20 ! x 2 #
4
4
3
y# x
2
&
2
5x2 80 # 5x2
" 20 #
"
4
4
3x $80 # 5x2
3x $80 # 5x2
#
and y2 " ! !
.
2
2
2
2
−12
12
−8
$80 # 5x2
3
y"! x ±
2
2
17. 17x2 # 32xy ! 7y2 " 75
16. x2 ! 4xy # 2y2 " 8
A " 1, B " !4, C " 2
cot 2% "
cot 2% "
A!C 1!2 1
"
"
B
!4
4
A ! C 17 # 7 24 3
"
"
" ⇒ % + 26.57&
B
32
32 4
Solve for y in terms of x by completing the square.
!7y2 # 32xy " !17x2 # 75
1
1
"
tan 2% 4
y2 !
tan 2% " 4
2% + 75.96&
y2 !
% + 37.98&
32
256 2 119 2 525 256 2
xy #
x "
x !
#
x
7
49
49
49
49
%y ! 7 x&
16
To graph conic with a graphing calculator, we need
to solve for y in terms of x.
x2 ! 4xy # 2y2 " 8
y2
! 2xy #
x2
x2
" 4 ! # x2
2
© Houghton Mifflin Company. All rights reserved.
! y ! x"2 " 4 #
x2
2
$4 # x2
x
y " x ± $4 #
2
x
Graph y " x # $4 # and
2
x
y " x ! $4 # .
2
Graph y1 "
2
y!x"±
y2 "
2
2
1
2
2
6
−9
9
−6
32
17
75
xy " x2 !
7
7
7
2
"
375x2 ! 525
49
16
x±
7
y"
16x ± 5$15x2 ! 21
7
16x # 5$15x2 ! 21
and
7
16x ! 5$15x2 ! 21
.
7
6
−9
9
−6
$
375x2 ! 525
49
y"
Chapter 9
Topics in Analytic Geometry
19. 32x 2 # 48xy # 8y 2 " 50
18. 40x2 # 36xy # 25y2 " 52
A " 40, B " 36, C " 25
cot 2% "
cot 2% "
A ! C 40 ! 25
5
"
"
B
36
12
1
5
"
tan 2% 12
tan 2% "
A ! C 32 ! 8 1
"
" ⇒ % + 31.72&
B
48
2
Solve for y in terms of x:
8y 2 # 48xy " !32x 2 # 50
2
y 2 # 6xy " !4x 2 #
12
5
−3
3
−2
2% + 67.38&
y 2 # 6xy # 9x 2 " !4x 2 #
% + 33.69&
! y # 3x"2 " 5x 2 #
Solve for y in terms of x by completing the square:
y2 #
y2 #
36
52 40 2
xy "
! x
25
25 25
36
324 2 52 40 2 324 2
xy #
x "
! x #
x
25
625
25 25
625
%
18
y# x
25
y#
&
2
1300 ! 676x2
"
625
18
x"±
25
y"
Graph y1 "
y2 "
$
1300 ! 676x2
625
Graph y1 " !3x #
25
# 9x 2
4
25 20x2 # 25
"
4
4
y " !3x ±
25y2 # 36xy " 52 ! 40x2
25
4
$20x2 # 25
2
$20x2 # 25
2
$20x2 # 25
y2 " !3x !
.
2
and
6
−9
9
−6
!18x ± $1300 ! 676x2
25
!18x # $1300 ! 676x2
and
25
!18x ! $1300 ! 676x2
.
25
20. 4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91
A " 4, B " !12, C " 9
cot 2% "
A!C 4!9
5
"
"
B
!12
12
1
5
"
tan 2% 12
tan 2% "
12
5
2% + 67.38&
% + 33.69&
—CONTINUED—
© Houghton Mifflin Company. All rights reserved.
816
Section 9.4
Rotation and Systems of Quadratic Equations
20. —CONTINUED—
Solve for y in terms of x with the Quadratic Formula:
4x2 ! 12xy # 9y2 # !4$13 ! 12"x ! !6$13 # 8"y " 91
9y2 ! !12x # 6$13 # 8"y # !4x2 # 4$13x ! 12x ! 91" " 0
a " 9, b " ! !12x # 6$13 # 8", c " 4x2 # 4$13x ! 12x ! 91
y"
y"
"
!b ± $b2 ! 4ac
2a
!12x # 6$13 # 8" ± $!12x # 6$13 # 8"2 ! 4!9"!4x2 # 4$13x ! 12x ! 91"
18
!12x # 6$13 # 8" ± $624x # 3808 # 96$13
Graph y1 "
y2 "
12x # 6$13 # 8 # $624x # 3808 # 96$13
and
18
−9
12x # 6$13 # 8 ! $624x # 3808 # 96$13
.
18
21. xy # 4 " 0
B2 ! 4AC " 1 ⇒ The graph is a hyperbola.
cot 2% "
A!C
" 0 ⇒ % " 45&
B
Matches graph (e).
23. !2x2 # 3xy # 2y2 # 3 " 0
B2 ! 4AC " !3"2 ! 4!!2"!2"
" 25 ⇒ The graph is a hyperbola.
cot 2% "
© Houghton Mifflin Company. All rights reserved.
18
18
A!C
4
" ! ⇒ % + !18.43&
B
3
Matches graph (f).
27
−6
22. x2 # 2xy # y2 " 0
!x # y"2 " 0
x#y"0
y " !x
The graph is a line. Matches graph (b).
24. x2 ! xy # 3y2 ! 5 " 0
A " 1, B " !1, C " 3
B2 ! 4AC " !!1"2 ! 4!1"!3" " !11
The graph is an ellipse or circle.
cot 2% "
A!C 1!3
"
" 2 ⇒ % + 13.28&
B
!1
Matches graph (a).
25. 3x2 # 2xy # y2 ! 10 " 0
B2 ! 4AC " !2"2 ! 4!3"!1"
cot 2% "
26. x2 ! 4xy # 4y2 # 10x ! 30 " 0
A " 1, B " !4, C " 4
" !8 ⇒ The graph is an ellipse or circle.
B2 ! 4AC " !!4"2 ! 4!1"!4" " 0
A!C
" 1 ⇒ % " 22.5&
B
The graph is a parabola.
Matches graph (d).
cot 2% "
A!C 1!4 3
"
" ⇒ % + 26.57&
B
!4
4
Matches graph (c).
817
Chapter 9
818
Topics in Analytic Geometry
27. 16x 2 ! 24xy # 9y 2 ! 30x ! 40y " 0
(a) B2 ! 4AC " !!24"2 ! 4!16"!9" " 0 ⇒ Parabola
(c)
6
(b) 9y 2 ! !24x # 40"y # !16x 2 ! 30x" " 0
"
!24x # 40" ± $!24x # 40"2 ! 4!9"!16x2 ! 30x"
2!9"
−4
−2
24x # 40 ± $3000x # 1600
18
28. (a) B2 ! 4AC " !!4"2 ! 4!1"!!2"
29. 15x2 ! 8xy # 7y2 ! 45 " 0
(a) B2 ! 4AC " !!8"2 ! 4!15"!7"
" 16 # 8 " 24
> 0 ⇒ Hyperbola
(b)
!2y 2
y"
"
(c)
! 4xy #
x2
" !356 ⇒ Ellipse or circle
!6"0
(b) 7y2 ! 8xy # !15x2 ! 45" " 0
4x ± $16x2 ! 4!!2"!x2 ! 6"
!4
y"
4x ± $24x2 ! 48
!4
"
(c)
8
−12
12
6
−4
31. x2 ! 6xy ! 5y2 # 4x ! 22 " 0
(a) B2 ! 4AC " !!6"2 ! 4!1"!!5"
" !24 < 0 ⇒ Ellipse or circle
" 56 ⇒ Hyperbola
(b) 5y 2 # !4x ! 4"y # !2x 2 # 3x ! 20" " 0
(c)
8x ± $1260 ! 356x2
14
−6
30. (a) B2 ! 4AC " 42 ! 4!2"!5"
"
8x ± $!!8x"2 ! 4!7"!15x2 ! 45"
14
4
−8
y"
8
(b) !5y2 ! 6xy # !x2 # 4x ! 22" " 0
!4 ! 4x" ± $!4x ! 4"2 ! 4!5"!2x 2 # 3x ! 20"
10
!4 ! 4x"
± $!24x2
y"
! 92x # 416
10
"
6
−10
(c)
8
6x ± $!!6x"2 ! 4!!5"!x2 # 4x ! 22"
!10
6x ± $56x2 # 80x ! 440
!10
6
−10
8
−6
−6
32. (a) B2 ! 4AC " !!60"2 ! 4!36"!25"
" 0 ⇒ Parabola
(c)
(b) 25y 2 # !9 ! 60x"y # 36x 2 " 0
y"
1
−10
2
"
−7
!60x ! 9" ± $!9 ! 60x"2 ! 100!36x 2"
50
60x ! 9 ± $!1080x # 81
50
© Houghton Mifflin Company. All rights reserved.
y"
Section 9.4
Rotation and Systems of Quadratic Equations
34. (a) B2 ! 4AC " 1 ! 4!1"!4"
33. x2 # 4xy # 4y2 ! 5x ! y ! 3 " 0
(a) B2 ! 4AC " 42 ! 4!1"!4" " 0 ⇒ Parabola
" !15 < 0 ⇒ Ellipse or circle
(b) 4y 2 # !x # 1"y # !x 2 # x ! 4" " 0
(b) 4y2 # !4x ! 1"y # !x2 ! 5x ! 3" " 0
y"
"
(c)
!1 ! 4x" ± $!4x ! 1"2 ! 4!4"!x2 ! 5x ! 3"
8
y"
1 ! 4x ± $72x # 49
8
"
! !x # 1" ± $!x # 1"2 ! 4!4"!x 2 # x ! 4"
8
!x ! 1 ± $!15x2 ! 14x # 65
8
(c)
3
−4
819
8
2
−3
3
−2
−5
35. y2 ! 16x2 " 0
36.
y
3
y2 " 16x2
x2 # y2 ! 2x # 6y # 10 " 0
!x2 ! 2x # 1" # ! y2 # 6y # 9" " !10 # 1 # 9
!x ! 1"2 # ! y # 3"2 " 0
y " ± 4x
Two intersecting lines
−3
−2
x
−1
1
2
Point at !1, !3"
3
y
4
3
2
1
x
−4 −3 −2 −1
1
−2
2
3
4
(1, −3)
−3
−4
38.
37. x2 # 2xy # y2 ! 1 " 0
!x # y"2 ! 1 " 0
x2 ! 10xy # y2 " 0
y2 ! 10xy # 25x2 " 25x2 ! x2
! y ! 5x"2 " 24x2
!x # y"2 " 1
y ! 5x " ± $24x2
© Houghton Mifflin Company. All rights reserved.
x # y " ±1
y " 5x ± 2$6x
y " !x ± 1
y " !5 ± 2$6 "x
y
Two parallel lines
3
Two lines
2
y
4
1
−3
−2
−1
3
1
−2
−3
2
3
x
2
1
−2 −1
−2
x
1
2
3
4
820
Chapter 9
Topics in Analytic Geometry
40.
39. x2 # y2 " 4
4x 2 # 9y 2 ! 36y " 0
x 2 # y 2 ! 27 " 0 ⇒ x 2 " 27 ! y 2
3x ! y2 " 0
4!27 ! y 2" # 9y 2 ! 36y " 0
Adding:
5y 2 ! 36y # 108 " 0
x2 # 3x ! 4 " 0
!x # 4"!x ! 1" " 0 ⇒ x " 1, !4
y"
For x " 1, y " ± $3.
x " !4 is impossible.
"
Solutions: !1, $3 ", !1, ! $3 "
36 ± $!864
10
No solution
For x " !8:
41. !4x2 ! y2 ! 16x # 24y ! 16 " 0
!4!64" ! y 2 ! 16!!8" # 24y ! 16 " 0
4x2 # y2 # 40x ! 24y # 208 " 0
24x
36 ± $362 ! 4!5"!108"
10
!y2 # 24y ! 144 " 0
# 192 " 0
y2 ! 24y # 144 " 0
24x " !192
! y ! 12"2 " 0
x " !8
⇒ y " 12
Solution: !!8, 12"
42.
x2 ! 4y2 ! 20x ! 64y ! 172 " 0 ⇒
!x ! 10"2 ! 4! y # 8"2 " 16
16x2 # 4y2 ! 320x # 64y # 1600 " 0 ⇒ 16!x ! 10"2 # 4! y # 8"2 " 256
17x2
!340x
# 1428 " 0
!17x ! 238"!x ! 6" " 0
When x " 6:
When x " 14:
62 ! 4y2 ! 20!6" ! 64y ! 172 " 0
142 ! 4y 2 ! 20!14" ! 64y ! 172 " 0
!4y2 ! 64y ! 256 " 0
4y 2 # 64y # 256 " 0
y2 # 16y # 64 " 0
y 2 # 16y # 64 " 0
! y # 8"2 " 0
! y # 8"2 " 0
y " !8
y " !8
Points of intersection: !6, !8", !14, !8"
43.
x2 ! y2 ! 12x # 16y ! 64 " 0
For x " 0:
x2 # y2 ! 12x ! 16y # 64 " 0
!y2 # 16y ! 64 " 0
2x2
! 24x
"0
x2 ! 12x " 0
x!x ! 12" " 0 ⇒ x " 0, 12
y2 ! 16y # 64 " 0
! y ! 8"2 " 0 ⇒ y " 8
For x " 12:
144 ! y2 ! 12!12" # 16y ! 64 " 0
!y2 # 16y ! 64 " 0 ⇒ y " 8
Solutions: !0, 8", !12, 8"
© Houghton Mifflin Company. All rights reserved.
x " 6 or x " 14
Section 9.4
Rotation and Systems of Quadratic Equations
x2 # 4y2 ! 2x ! 8y # 1 " 0 ⇒ !x ! 1"2 # 4! y ! 1"2 " 4
44.
# 2x ! 4y ! 1 " 0 ⇒ y " ! 14!x ! 1"2
!x2
4y2
"0
!12y
4y! y ! 3" " 0
y " 0 or y " 3
When y " 0:
x2
When y " 3:
# 4!0" ! 2x ! 8!0" # 1 " 0
!x2 # 2x ! 4!3" ! 1 " 0
x2 ! 2x # 1 " 0
x2 ! 2x # 13 " 0
2
!x ! 1"2 " 0
No real solution
x"1
Point of intersection: !1, 0"
45. !16x2 !
y2 # 24y ! 80 " 0
16x2 # 25y2
! 400 " 0
24y 2 # 24y ! 480 " 0
24! y # 5"! y ! 4" " 0
y " !5 or y " 4
When y " 4:
When y " !5:
16x2
# 25!!5" ! 400 " 0
2
16x2 " !225
16x2 # 25!4"2 ! 400 " 0
16x2 " 0
No real solution
x"0
The point of intersection is !0, 4". In standard form the equations are:
© Houghton Mifflin Company. All rights reserved.
x2 ! y ! 12"2
#
"1
4
64
x2
y2
#
"1
25 16
# 16y ! 128 " 0 ⇒ 16x2 ! ! y ! 8"2 " 64
46. 16x2 ! y2
y2 ! 48x ! 16y ! 32 " 0 ⇒ ! y ! 8"2 ! 48x " 96
16x2
! 48x
! 160 " 0
16!x2 ! 3x ! 10" " 0
!x ! 5"!x # 2" " 0
x " 5 or x " !2
When x " 5:
y2
When x " !2:
! 48!5" ! 16y ! 32 " 0
y2 ! 48!!2" ! 16y ! 32 " 0
y2 ! 16y ! 272 " 0
y2 ! 16y # 64 " 0
y " 8 ± 4$21
! y ! 8"2 " 0
y"8
Points of intersection: !5, 8 # 4$21 ", !5, 8 ! 4$21 ", !!2, 8"
821
822
Chapter 9
Topics in Analytic Geometry
48.
47. 2x 2 ! y 2 # 6 " 0
6x 2 # 3y 2 ! 12 " 0
2x # y " 0 ⇒ y " !2x
x#y!2"0 ⇒ y"2!x
2x 2 ! !!2x"2 # 6 " 0
6x 2 # 3!2 ! x"2 ! 12 " 0
!2x 2 # 6 " 0
6x 2 # 3!4 ! 4x # x 2" ! 12 " 0
x 2 " 3 ⇒ x " ± $3
9x 2 ! 12x " 0
Two solutions: !$3, !2$3 ", !! $3, 2$3 "
3x!3x ! 4" " 0
x"0 ⇒ y"2
x " 43 ⇒ y " 2 ! 43 " 23
Solutions: !0, 2", !43, 23 "
50.
49. 10x 2 ! 25y 2 ! 100x # 160 " 0
y 2 ! 2x # 16 " 0 ⇒ y 2 " 2x ! 16
4x 2 ! y 2 ! 8x # 6y ! 9 " 0
2x 2 ! 3y 2 # 4x # 18y ! 43 " 0
From Equation 1:
10x2 ! 25!2x ! 16" ! 100x # 160 " 0
y 2 ! 6y " 4x 2 ! 8x ! 9
10x 2 ! 150x # 560 " 0
x 2 ! 15x # 56 " 0
3y 2 ! 18y " 12x 2 ! 24x ! 27
!x ! 8"!x ! 7" " 0
In Equation 2:
2x 2 # 4x ! 43 ! !3y 2 ! 18y" " 0
x " 8 ⇒ y 2 " 0 ⇒ !8, 0"
2x 2 # 4x ! 43 ! !12x 2 ! 24x ! 27" " 0
x " 7 ⇒ y 2 " !2 impossible
!10x 2 # 28x ! 16 " 0
One solution: !8, 0"
5x 2 ! 14x # 8 " 0
!x ! 2"!5x ! 4" " 0
x " 2 ⇒ y 2 ! 6y " !9
⇒ ! y ! 3"2 " 0 ⇒ y " 3
x " 45 ⇒ y 2 ! 6y " ! 321
25
No solution
51.
xy # x ! 2y # 3 " 0 ⇒ y "
x2
x2 # 4
#
%
4y2
!x ! 3
x!2
!9"0
!x ! 3
x!2
&
2
"9
x2!x ! 2"2 # 4!!x ! 3"2 " 9!x ! 2"2
x2!x2 ! 4x # 4" # 4!x2 # 6x # 9" " 9!x2 ! 4x # 4"
x4 ! 4x3 # 4x2 # 4x2 # 24x # 36 " 9x2 ! 36x # 36
x 4 ! 4x3 ! x2 # 60x " 0
x(x # 3)!x2 ! 7x # 20" " 0
x " 0 or x " !3
Note: x2 ! 7x # 20 " 0 has no real solution.
When x " 0: y "
!0 ! 3 3
"
0!2
2
When x " !3: y "
! !!3" ! 3
"0
!3 ! 2
The points of intersection are !0, 32 ", !!3, 0".
© Houghton Mifflin Company. All rights reserved.
Solution: !2, 3"
Section 9.4
Rotation and Systems of Quadratic Equations
823
52. 5x2 ! 2xy # 5y2 ! 12 " 0
x#y!1"0 ⇒ y"1!x
5x2 ! 2x!1 ! x" # 5!1 ! x"2 ! 12 " 0
5x2 ! 2x # 2x2 # 5!1 ! 2x # x2" ! 12 " 0
5x2 ! 2x # 2x2 # 5 ! 10x # 5x2 ! 12 " 0
12x2 ! 12x ! 7 " 0
x"
3 ± $30
6
When x "
3 # $30
3 # $30
3 ! $30
: y " 1!
"
6
6
6
When x "
3 ! $30
3 ! $30 3 # $30
: y " 1!
"
6
6
6
Points of intersection:
%6!3 # $30 ", 6!3 ! $30 "&, %6!3 ! $30 ", 6!3 # $30 "&
1
1
1
53. True. B2 ! 4AC " 1 ! 4k
If k <
55. g!x" "
1
4,
then
B2
1
54. False. See Example 2. However, A # C " A$ # C$.
! 4AC > 0.
2
2!x
y
56. f !x" "
4
3
Asymptotes:
x " 2, y " 0
Intercepts: !0, 1"
Intercept: !0, 0"
2
(0, 1)
x
−2 −1
2x
4
" !2 #
2!x
2!x
1
3
−2
y
6
Asymptotes:
x " 2, y " !2
4
2
(0, 0)
−3
x
−4
4
6
8
−4
© Houghton Mifflin Company. All rights reserved.
−6
57. h!t" "
t2
4
" !t ! 2 #
2!t
2!t
58. g!s" "
Slant asymptote: y " !t ! 2
Vertical asymptote: t " 2
Intercept:
y
Intercept: !0, 0"
−10
(0, 0)
t
−5
5
−5
−10
−15
10
15
y
4
3
%0, 12&
Asymptotes:
s " ± 2, y " 0
10
5
2
4 ! s2
2
(0, 12 )
1
−4
−1
−2
−3
−4
1
4
s
Chapter 9
824
Topics in Analytic Geometry
59. (a) AB "
)12
!3
5
(b) BA "
)05
6
!1
)12
!3
5
61. (a) AB " '4
!2
(c) A2 "
*)05
6
!15
"
!1
25
* )
*)12
!3
12 30
"
5
3 !20
*)12
*
9
7
)10
60. (a) AB "
* )
*
(b) BA "
* )
*
(c) A2 "
!3
!5 !18
"
5
12 19
*)!33
5
!2
)!33
)10
*)10
2
8
*)10
* )
*
* )
*
2
!12 42
"
8
6 !16
5
3 11
"
!2
!3 !31
* )
5
!2
*
5
1
"
!2
0
!5
4
) *
3
5( !4 " '12 # 8 # 25( " '45(
5
) *
3
(b) BA " !4 '4
5
)
12
5( " !16
20
!2
15
!20
25
!6
8
!10
*
(c) A2 does not exist.
)
)
)
*
*
*
8 !10
2
27 20
6
!13 20 !13
(b) BA "
,
!2 !10
19
5
!2 20
65. g!x" " $4 ! x 2
,
y
10
10
8
8
6
4
4
2
−8
−6
−4
−2
x
2
−2
4
−2
x
−2
2
4
8
6
x
−2
−1
−1
1
2
4
3
2
−2
−2
−3
68. h!t " " 12!t # 4"3
x
2
−2
−3 − 2 −1
1
1
2
x
−2
−3
−4
−10 −8 −6 − 4 −2
2
x
6 8 10
70. f !t" " !2-t. # 3
y
4
1 2
5 6 7
3
x
2
1
− 4 − 3 −2 − 1
−2
−3
−4
10
y
2
−2
8
3
2
1
3
− 4 −3 −2 −1
−1
6
−4
y
−6
4
69. f !t" " -t ! 5. # 1
4
10
16
14
12
10
8
6
4
2
10
1
8
y
y
3
6
67. h!t " " ! !t ! 2"3 # 3
66. g!x" " $3x ! 2
y
−3
,
64. f !x" " x ! 4 # 1
y
0
25
15
!9 !14
2
9
9 !1
!2
(c) A2 " 16
4
,
63. f !x" " x # 3
−7
−3
1
2
3
4
x
© Houghton Mifflin Company. All rights reserved.
62. (a) AB "
Section 9.5
! 12"8#"12# sin 110"
! 12"25#"16# sin 70"
! 45.11
! 187.94
73. s ! 12"11 $ 18 $ 10# ! 39
2
1
85
74. s ! 2"23 $ 35 $ 27# ! 2
Area ! $s"s # a#"s # b#"s # c#
■
■
■
Area ! $s"s # a#"s # b#"s # c#
17 3 19
! $39
2 " 2 #"2 #" 2 #
! $ 2 " 2 #" 2 #" 2 #
! 48.60
! 310.39
Section 9.5
825
1
72. Area ! 2ac sin B
71. Area ! 12ab sin C
■
Parametric Equations
85 39
15
31
Parametric Equations
If f and g are continuous functions of t on an interval I, then the set of ordered pairs "f"t#, g"t## is a plane
curve C. The equations x ! f "t# and y ! g"t# are parametric equations for C and t is the parameter.
You should be able to graph plane curves with your graphing utility.
To eliminate the parameter:
Solve for t in one equation and substitute into the second equation.
You should be able to find the parametric equations for a graph.
Vocabulary Check
1. plane curve, parametric equations, parameter
2. orientation
3. eliminating, parameter
2. x ! t 2
© Houghton Mifflin Company. All rights reserved.
1. x ! t
3. x ! $t
y!t$2
y!t#2 ⇒ t!y$2
y!t
y ! x $ 2, line
x ! " y $ 2#
y ! x2, parabola, x ≥ 0
Matches (c).
Parabola opening to the right
Matches (d).
4. x !
1
1
⇒ t!
t
x
y!t$2 ⇒ y!
Matches (a).
2
5. x ! ln t ⇔ t ! ex
1
$2
x
1
y! t#2
2
Matches (b).
6. x ! #2$t ⇒ t !
&#2'
x
2
y ! et
2
y ! ex %4
1
y ! ex # 2
2
Exponential curve on x ≤ 0
Matches (f).
Matches (e).
!
x2
4
Chapter 9
826
Topics in Analytic Geometry
7. x ! $t, y ! 2 # t
(a)
(c)
t
0
1
2
3
4
x
0
1
$2
$3
2
y
2
1
0
#1
#2
3
−4
5
−3
(b) Graph by hand.
y
Note: x ≥ 0
(d) y ! 2 # t ! 2 # x 2,
2
Parabola
1
−2
1
1
In part (c), x ≥ 0.
x
−1
y
2
−2
−1
x
−1
1
2
−1
−2
−2
8. x ! 4 cos2 %, y ! 4 sin %
(a)
&
2
%
#
x
0
y
#4
(b)
0
&
4
&
2
2
4
2
0
#2$2
0
2$2
4
#
&
4
(c)
6
−8
10
−6
(d) 4x $ y2 ! 16 cos2 % $ 16 sin2 % ! 16
y
x
y2
$
! 1, parabola
4
16
5
4
3
2
1
− 5 − 4 −3 − 2 − 1
1 2 3
5
x
−2
−3
−4
−5
The graph is an entire
parabola rather than
just the right portion.
y
5
3
2
1
− 5 − 4 − 3 −2 − 1
1 2 3
5
x
−2
−3
9. The graph opens upward, contains "1, 0#, and is
oriented left to right. Matches (b).
10. The orientation of the graph is clockwise and the
center is "2, 3#. Matches (c).
1
12. x ! t, y ! t
2
11. x ! t, y ! #4t
y ! #4x
1
y! x
2
y
or
2
x # 2y ! 0
y
6
−2
4
x
−1
1
2
2
−1
−6
−2
−4
x
−2
−4
−6
2
4
6
© Houghton Mifflin Company. All rights reserved.
−5
Section 9.5
x$3
3
t!
y!2$3
&
1
15. x ! t, y ! t 2
4
14. x ! 3 # 2t, y ! 2 $ 3t
13. x ! 3t # 3, y ! 2t $ 1
'
x$3
y!2
$1
3
Parametric Equations
&
3#x
2
'
y ! "4x#2
y ! 16x2
3x $ 2y # 13 ! 0
y
y
5
2
y! x$3
3
6
4
y
2
5
4
−4
2
x
−2
2
−2
6
4
8
−3
−2
−1
x
1
−1
3
2
−4
1
−6
−4 −3 −2 −1
−1
1
x
2
−2
−3
18. x ! $t
17. x ! t $ 2, y ! t 2
16. x ! t, y ! t 3
y ! x3
y
t!x#2
y!1#t
y ! "x # 2#2
y ! 1 # x2, x ≥ 0
6
y
y
4
2
2
−6
−4
−2
x
−2
2
6
4
−6
4
3
−4
2
−6
1
−2 −1
−1
(
© Houghton Mifflin Company. All rights reserved.
(
!
3
4
5
(
(
( (
Eliminating the parameter t, t ! y # 2 and
x
#2
2
1
x#4
2
(
( (
! (" y # 2# # 1(
! (y # 3(.
x! t#1
(
y
10
10
8
8
6
6
4
4
2
x
−2
2
4
6
8
10
−10
y
−2
8
−8
x
6
6
y!t$2
x
⇒ y! t#2
2
!
2
4
20. x ! t # 1
19. x ! 2t, y ! t # 2
t!
1
(
(
x
2
−2
10
−2
x
−2
2
6
8
10
827
828
Chapter 9
Topics in Analytic Geometry
21. x ! 2 cos %, y ! 3 sin %
22. x ! cos %, y ! 4 sin %
y
y
5
4
4
2
1
−4 −3
−1
−1
1
3
x
4
− 5 −4 − 3 − 2
2 3 4 5
x
−2
−4
−5
−4
2
! cos2 %,
&3y '
2
! sin2 %
x2 $
x 2 y2
$ ! cos2 % $ sin2 % ! 1
4
9
x 2 y2
$ ! 1,
4
9
23. x ! e#t ⇒
&'
y!
x2 $
2
1
! et
x
y2
! 1,
16
y ! e t ⇒ y 2 ! e2t
y 2 ! x, y > 0; y ! $x, x > 0
3
y
1
, x > 0, y > 0
x3
5
4
3
2
10
9
8
7
6
5
4
3
2
1
1
x
1
2
3
4
5
x
−2 −1
1 2
3 4 5 6 7 8
26. x ! ln 2t ⇒ e x ! 2t ⇒ t ! 12 e x
25. x ! t3 ⇒ x1%3 ! t
y ! 3 ln t ⇒ y ! ln t3
y ! 2t 2 ! 2"12 e x# ! 12 e 2x
2
y ! ln"x1%3#3
y
6
y ! ln x
5
y
4
5
4
3
2
1
−2
−3
−4
−5
ellipse
24. x ! e 2t
y
−2 −1
! cos2 % $ sin2 % ! 1
ellipse
y ! e 3t ⇒ y ! "e t#3
1
y!
x
&4y '
3
2
1
x
2 3 4 5 6 7 8
−3
−2
−1
1
2
3
x
© Houghton Mifflin Company. All rights reserved.
&2x '
Section 9.5
27. x ! 4 $ 3 cos %, y ! #2 $ sin %
Parametric Equations
28. x ! 4 $ 3 cos %, y ! #2 $ 2 sin %
1
2
−1
8
−2
−5
10
−6
30. x ! sec %
29. x ! 4 sec %, y ! 2 tan %
−12
4
y ! tan %
8
−6
6
12
−4
−8
31. x !
t
2
8
−12
y ! ln"t 2 $ 1#
32. x ! 10 # 0.01e t
35
y ! 0.4t 2
12
−40
−8
(a) Domain: # ' < x <
(b) Domain: #1 ≤ x ≤ 1
'
Orientation: Depends on %
Orientation: Left to right
(c) Domain: 0 < x <
(d) Domain: 0 < x <
'
'
Orientation: Left to right
Orientation: Right to left
34. Each curve represents a portion of the line 2y $ x # 8 ! 0.
3
(b) x ! 2 $
t, # ' < x <
(a) x ! 2$t, x ≥ 0
'
x
2
x
y ! 4 # $t ! 4 # , y ≤ 4
2
3 t ! 4 #
y ! 4 #$
Orientation: Left to right
Orientation: Left to right
(c) x ! 2"t $ 1#, # ' < x <
y!3#t!3#
'
&x #2 2' ! 4 # 2x
Orientation: Left to right
20
−5
33. By eliminating the parameters in (a)–(d), we get y ! 2x $ 1. They differ from each other
in restricted domain and in orientation.
© Houghton Mifflin Company. All rights reserved.
829
(d) x ! #2t 2, x ≤ 0
y ! 4 $ t2 ! 4 #
x
2
Orientation: Left to right for t ≤ 0
Right to left for t > 0
830
Chapter 9
35. t !
Topics in Analytic Geometry
"x # x1#
"x2 # x1#
36. x ! h $ r cos %
y ! k $ r sin %
x # x1
y ! y1 $
" y # y1#
x2 # x1 2
&
'
⇒ y # y1 !
"x # h#
y#k
! cos %,
! sin %
r
r
y2 # y1
"x # x1#
2 # x1
&x
'
"x # h# 2 " y # k# 2
$
!1
r2
r2
cos 2 % $ sin 2 % !
"x # h# 2 $ " y # k# 2 ! r 2
37. x ! h $ a cos %
38. x ! h $ a sec %
y ! k $ b sin %
y ! k $ b tan %
x#h
y#k
! cos %,
! sin %
a
b
x#h
! sec %,
a
"x # h#2 " y # k#2
$
!1
a2
b2
sec 2 % # tan 2 % !
39. x ! x1 $ t"x2 # x1# ! 1 $ t"6 # 1# ! 1 $ 5t
"x # h# 2 "y # k# 2
#
!1
a2
b2
40. x ! h $ r cos % ! 2 $ 4 cos %
y ! y1 $ t" y2 # y1# ! 4 $ t"#3 # 4# ! 4 # 7t
y ! k $ r sin % ! 5 $ 4 sin %
42. a ! 1, c ! 2, and b ! $c2 # a2 ! $3.
41. a ! 5, c ! 4, and b ! $a2 # c2 ! 3.
The center is "0, 0#, so h ! 0 and k ! 0.
The center is "0, 0#, so h ! 0 and k ! 0.
x2
y2
so x ! 5 cos % and
2 $ 2,
5
3
y ! 3 sin %. This solution is not unique.
sec2 % # tan2 % ! 1 !
cos2 % $ sin2 % ! 1 !
y2 x2
# , so y ! sec % and
1
3
x ! $3 tan %.
45. y !
Answers will vary.
Answers will vary.
x ! t, y ! 5t # 3
x ! t, y ! 4 # 7t
1
x ! t, y ! t # 3
5
x ! 2t, y ! 4 # 14t
1
x
Sample answers:
x ! t, y !
1
t
x ! t 3, y !
46. y !
1
2x
47. y ! 6x2 # 5
x ! t, y !
1
x ! t, y !
2t
x ! 2t, y !
6t 2
#5
x ! 2t, y ! 24t 2 # 5
1
4t
49. x ! 2 cot %, y ! 2 sin2 %
4
−6
50. x !
6
−4
1
t3
48. y ! x3 $ 2x
Sample answers:
Sample answers:
© Houghton Mifflin Company. All rights reserved.
44. y ! 4 # 7x
43. y ! 5x # 3
y#k
! tan %
b
Sample answers:
x ! t, y ! t 3 $ 2t
1
t3
x ! t, y ! $ t
2
8
3t
3t 2
,
y
!
1 $ t3
1 $ t3
4
−6
6
−4
Section 9.5
51. Matches (b).
52. Matches (c).
55. x ! "v0 cos %#t, y ! h $ "v0 sin %#t # 16t2
(a) 100 miles%hour !
53. Matches (d).
831
54. Matches (a).
(c) % ! 23"
x ! "146.67 cos 23"#t ! 135.0t
100 mi(hr ) 5280 ft(mi
3600 sec(hr
y ! 3 $ "146.67 sin 23"#t # 16t2
! 146.67 ft%sec
! 3 $ 57.3t # 16t2
x ! "146.67 cos %#t
y ! 3 $ "146.67 sin %#t #
Parametric Equations
60
16t2
(b) % ! 15"
x ! "146.67 cos 15"#t ! 141.7t
0
y ! 3 $ "146.67 sin 15"#t # 16t2
! 3 $ 38.0t #
Yes, it is a home run because y > 10 when
x ! 400.
16t2
(d) % ! 19.4" is the minimum angle.
30
0
500
0
450
0
It is not a home run because y < 10 when x ! 400.
56. (a) x ! "v0 cos %#t ! "105 cos 40"#t
(b)
80
y ! h $ "v0 sin %#t # 16t 2
! 2.5 $ "105 sin 40"#t # 16t 2
(c) The horizontal distance is approximately
342.25 feet.
© Houghton Mifflin Company. All rights reserved.
(d) You could use the Quadratic Formula to find
the zeros of y ! #16t 2 $ "105 sin 40"#t $ 2.5.
The larger zero, 4.255, gives x ! 342.25 feet.
57. (a) x ! "cos 35"#v0 t
400
0
The maximum height is approximately
73.68 feet, when t ! 2.109 seconds.
(b) If the ball is caught at time t1, then:
y ! 7 $ "sin 35"#v0 t # 16t 2
(c)
0
90 ! "cos 35"#v0 t1
4 ! 7 $ "sin 35"#v0t1 # 16t12.
24
v0 t1 !
0
90
0
Maximum height ! 22 feet
(d) From part (b), t1 ! 2.03 seconds.
90
90
⇒ #3 ! "sin 35"#
# 16t12
cos 35"
cos 35"
⇒ 16t12 ! 90 tan 35" $ 3
⇒ t1 ! 2.03 seconds
⇒ v0 !
90
! 54.09 ft(sec
t1 cos 35"
832
Chapter 9
Topics in Analytic Geometry
58. (a) x ! "v0 cos %#t ! "85 cos 50"#t
y ! h $ "v0 sin %#t # 16t 2 ! "85 sin 50"#t # 16t 2
(b)
80
0
240
0
The maximum height is approximately 66.25 feet when t ! 2.035 seconds.
(c) The horizontal distance is approximately 222.35 feet.
(d) You could solve the equation y ! "85 sin 50"#t # 16t 2 ! 0 for t ! 4.0696.
Then, x ! 222.35 feet.
59. True
first set
x!t
60. False. The graph of x ! t 2, y ! t 2 represents the
portion of the line y ! x in the first quadrant.
y ! t 2 $ 1 ! x2 $ 1
x ! 3t
second set
y ! 9t 2 $ 1 ! "3t#2 $ 1 ! x2 $ 1
61. False. For example, x ! t 2 and y ! t does not
represent y as a function of x.
62. False. The equations represent a line.
63. Sample answer: x ! cos %
64. The graph is the same, but the orientation is
reversed.
65. f "#x# !
4"#x#2
4x2
! 2
! f "x#
2
"#x# $ 1 x $ 1
Symmetric about the y-axis
Even function
67. y ! e x * e#x; e#x * #e x
No symmetry
Neither even nor odd
66. f "x# ! $x, x ≥ 0
No symmetry
Neither even nor odd
68. "x # 2#2 ! y $ 4
y ! x2 # 4x
No symmetry
Neither even nor odd
© Houghton Mifflin Company. All rights reserved.
y ! #2 sin %
Section 9.6
Section 9.6
Polar Coordinates
Polar Coordinates
■
In polar coordinates you do not have unique representation of points. The point !r, !" can be represented
by !r, ! ± 2n%" or by !$r, ! ± !2n # 1"%" where n is any integer. The pole is represented by !0, !" where
! is any angle.
To convert from polar coordinates to rectangular coordinates, use the following relationships.
■
x " r cos !
y " r sin !
To convert from rectangular coordinates to polar coordinates, use the following relationships.
■
r " ± $x2 # y2
tan ! " y#x
If ! is in the same quadrant as the point !x, y", then r is positive. If ! is in the opposite quadrant as the point
!x, y", then r is negative.
You should be able to convert rectangular equations to polar form and vice versa.
■
Vocabulary Check
1. pole
2. directed distance, directed angle
1. Polar coordinates:
x " 4 cos
y " 4 sin
%4, 2 &
%
3. polar
2. Polar coordinates:
%2& " 0
%
x " 4 cos
%4, 2 &
3%
% 2 & " 0, y " 4 sin% 2 & " $4
3%
3%
Rectangular coordinates: !0, $4"
%2& " 4
%
© Houghton Mifflin Company. All rights reserved.
Rectangular coordinates: !0, 4"
3. Polar coordinates:
y " $1 sin
5%
%4&"
$2
%4&"
$2
x " $1 cos
5%
5%
%
%$1, 4 &
4. Polar coordinates: 2, $
&
% %4 & " 2% 22& " $2
x " 2 cos $
2
$
% %4 & " 2%$ 22& " $ $2
y " 2 sin $
2
Rectangular coordinates:
%
4
% 22, 22 &
$
$
$
Rectangular coordinates: !$2, $ $2 "
833
Chapter 9
834
5.
Topics in Analytic Geometry
6.
π
2
( (
( (
2, 3π
4
3, 5π
6
1
2
π
2
0
3
1
2
0
3
Three additional representations:
Three additional points:
%3, 56% $ 2%& " %3, $ 76%&
%2, $54 %&, %$2, 74%&, %$2, $4%&
%$3, 56% # %& " %$3, 116%&
%$3, 56% $ %& " %$3, $ %6 &
7.
8.
π
2
1
2
π
2
0
3
1
2
(−1, − π3 (
3
4
0
)−3, − 76π )
Three additional representations:
Three additional points:
%$1, $ %3 # 2%& " %$1, 53%&
%$3, 56%&, %3, 116%&, %3, $ %6 &
%1, $ %3 # %& " %1, 23%&
9.
)
3, 5π
6
)
10.
π
2
1
2
3
0
π
2
(5
1
3
5
2, − 11π
6
7
9
)
0
Three additional representations:
Three additional points:
%$3, $ 76%&, %$ $3, $ %6 &, %$ $3, 116%&
%5$2, %6 &, %$5$2, $ 56%&, %$5$2, 76%&
© Houghton Mifflin Company. All rights reserved.
%1, $ %3 $ %& " %1, $ 43%&
Section 9.6
11.
π
2
12.
) 32 , − 32π )
1
3
2
Polar Coordinates
π
2
(0, − 4π (
0
1
2
0
3
%0, $4%& is the origin.
Three additional representations:
%32, %2 &, %$ 32, 32%&, %$ 32, $ %2 &
Three additional points:
%0, 34%&, %0, $54 %&, %0, 74%&
!Any angle will do since r " 0."
13. Polar coordinates:
%4, $ 3 &
%
14. Polar coordinates:
%
&
x " 2 cos
$3
7%
"2 $
" $ $3
6
2
% 3 & " $2$3
y " 2 sin
7%
1
" 2 $ " $1
6
2
y " 4 sin $
%
%
Rectangular coordinates: !2, $2$3 "
2
x " $1 cos
%
%
6
4
15. Polar coordinates:
y " $1 sin
% &
Rectangular coordinates: !$ $3, $1"
π
2
π
2
© Houghton Mifflin Company. All rights reserved.
%2, 76%&
% 3& " 2
x " 4 cos $
0
%$1,
1
$3%
4
&
2
2%
%
" 3,
3
3
&
% &
&
%
$3
2%
3$3
" $3 $
"
3
2
2
y " $3 sin $
$2 $2
2
&
Rectangular coordinates:
π
2
π
2
2
3
0
& % &
%
&
$2
$3%
"
4
2
%2
%
2%
1
3
" $3 $ "
3
2
2
x " $3 cos $
,
0
%
&
Rectangular coordinates:
3
16. Polar coordinates: $3, $
$2
$3%
"
4
2
1
835
1
2
3
0
&
%32, 3 2 3 &
$
Chapter 9
Topics in Analytic Geometry
17. Polar coordinates:
%0, $ 6 &
7%
%
7%
"0
6
%
7%
"0
6
x " 0 cos $
y " 0 sin $
%
(origin!)
18. Polar coordinates: 0,
&
x " 0 cos
5%
"0
4
&
y " 0 sin
5%
"0
4
5%
4
& (origin!)
Rectangular coordinates: !0, 0"
Rectangular coordinates: !0, 0"
π
2
π
2
1
2
3
0
1
2
3
0
19. Polar coordinates: !$2, 2.36"
20. Polar coordinates: !$3, $1.57"
x " $2 cos!2.36" ' $1.004
x " $3 cos!$1.57" ' $0.0024
y " $2 sin!2.36" ' 0.996
y " $3 sin!$1.57" ' 3.000
Rectangular coordinates: !$1.004, 0.996"
Rectangular coordinates: !$0.0024, 3"
π
2
π
2
1
%
21. !r, !" " 2,
2%
9
2
3
0
1
%
& ⇒ !x, y" " !1.53, 1.29"
22. !r, !" " 4,
11%
9
2
3
0
& ⇒ !x, y" " !$3.06, $2.57"
23. !r, !" " !$4.5, 1.3" ⇒ !x, y" " !$1.204, $4.336"
24. !r, !" " !8.25, 3.5" ⇒ !x, y" " !$7.726, $2.894"
25. !r, !" " !2.5, 1.58" ⇒ !x, y" " !$0.02, 2.50"
26. !r, !" " !5.4, 2.85" ⇒ !x, y" " !$5.17, 1.55"
27. !r, !" " !$4.1, $0.5" ⇒ !x, y" " !$3.60, 1.97"
28. !r, !" " !8.2, $3.2" ⇒ !x, y" " !$8.19, 0.48"
29. Rectangular coordinates: !$7, 0"
r " 7, tan ! " 0, ! " 0
Polar coordinates: !7, %", !$7, 0"
y
5
4
3
2
1
x
−9 −8 −7 −6 −5 −4 −3 −2 −1
−2
−3
−4
−5
1
© Houghton Mifflin Company. All rights reserved.
836
Section 9.6
30. Rectangular coordinates: !0, $5"
r " 5, tan ! undefined, ! "
Polar coordinates:
31. Rectangular coordinates: !1, 1"
%
2
r " $2, tan ! " 1, ! "
%5, 2 &, %$5, 2 &
3%
%
Polar coordinates:
y
−3
−2
−1
%$2, 4 &, %$ $2, 4 &
%
x
1
−1
2
1
−3
−4
−2
−1
x
−2
−6
−3
Polar coordinates:
%
4
%3$2, 4 &, %$3$2, 4 &
5%
%
Polar coordinates:
5%
1
−1
1
2
−3
−2
−1
x
−3
−2
−3
2
3
35. !x, y" " !6, 9"
r " $62 # 92 " $117 ' 10.8
r " $3 # 1 " 2
$1
11%
,!"
$3
6
Polar coordinates:
1
−1
−4
34. Rectangular coordinates: !$3, $1"
%2,
tan ! "
&%
11%
5%
, $2,
6
6
&
9 3
" ⇒ ! ' 0.983
6 2
Polar coordinates: !10.8, 0.983", !$10.8, 4.124"
y
y
12
2
9
1
1
2
3
4
5
6
6
x
3
−2
−3
−4
−5
−6
%
2
x
−2
© Houghton Mifflin Company. All rights reserved.
%$6, 4 &, %$ $6, 4 &
3
1
−2 −1
−1
%
4
y
2
tan ! "
3
r " $3 # 3 " $6, tan ! " 1, ! "
y
−1
2
33. Rectangular coordinates: !$ $3, $ $3"
32. Rectangular coordinates: !$3, $3"
r " 3$2, tan ! " 1, ! "
1
−1
−5
−2
5%
3
3
2
−3
−3
%
4
y
−2
−4
Polar Coordinates
−3
3
−3
6
9
12
x
837
838
Chapter 9
Topics in Analytic Geometry
36. Rectangular coordinates: !5, 12"
37. !x, y" " !3, $2" ⇒ r " $32 # !$2"2 " $13
! " arctan!$ 23 " ' $0.588
r " $25 # 144 " 13, tan ! " 12
5 , ! ' 1.176
!r, !" ' !$13, $0.588"
Polar coordinates: !13, 1.176", !$13, 4.318"
y
12
10
8
6
4
2
x
−6 −4 −2
−2
2
4
6
8
39. !x, y" " !$3, 2" ⇒ r " $3 # 22 " $7
38. !x, y" " !$5, 2" ⇒ !r, !" " !5.39, 2.76"
! " arctan
% 23& ' 0.857
$
!r, !" ' !$7, 0.857"
% 4 & ' !6.0, 0.785"
40. !x, y" " !3$2, 3$2" ⇒ !r, !" " 6,
% 2, 3 & ⇒
5 4
$%52& # %43&
2
r"
! " arctan
!r, !" '
2
"
17
6
%4, 2& ⇒ !r, !" " !2.30, 0.71"
7 3
%5#2& ' 0.490
4#3
% 6 , 0.490&
17
43. x2 # y 2 " 9
44. x2 # y 2 " 16
r2 " 9
r 2 " 16
r"3
r"4
46.
42. !x, y" "
y"x
47.
r sin ! " r cos !
45.
r sin ! " 4
r " 4 csc !
48.
x"8
x"a
r cos ! " a
r cos ! " 8
sin ! " cos !
y"4
r " a sec !
r " 8 sec !
tan ! " 1
!"
49.
%
4
3x $ 6y # 2 " 0
50.
4x # 7y $ 2 " 0
3r cos ! $ 6r sin ! " $2
4r cos ! # 7r sin ! $ 2 " 0
r!3 cos ! $ 6 sin !" " $2
r!4 cos ! # 7 sin !" " 2
r"
2
6 sin ! $ 3 cos !
r"
2
4 cos ! # 7 sin !
© Houghton Mifflin Company. All rights reserved.
41. !x, y" "
%
Section 9.6
51. xy " 4
52.
!r cos !"!r sin !" " 4
Polar Coordinates
839
2xy " 1
2r cos !
& r sin ! " 1
2r 2 " sec ! csc !
r 2 cos ! sin ! " 4
r 2 " 12 sec ! csc ! " csc!2!"
r 2!2 cos ! sin !" " 8
r 2 sin 2! " 8
r 2 " 8 csc 2!
53. !x 2 # y 2"2 " 9!x 2 $ y 2"
54.
y2 $ 8x $ 16 " 0
!r 2"2 " 9!r 2 cos2 ! $ r 2 sin2 !"
r2 sin2 ! $ 8r cos ! $ 16 " 0
r2!1 $ cos2 !" $ 8r cos ! $ 16 " 0
r 2 " 9!cos2 ! $ sin2 !"
r 2 " 9 cos!2!"
r2 cos2 ! # 8r cos ! # 16 " r2
!r cos ! # 4"2 " r2
r " ± !r cos ! # 4"
r"
or r "
$4
1 # cos !
55. x 2 # y 2 $ 6x " 0
56. x2 # y2 $ 8y " 0
r 2 $ 6r cos ! " 0
r2 $ 8r sin ! " 0
r2 $ 2ar cos ! " 0
r !r $ 8 sin !" " 0
r!r $ 2a cos !" " 0
r 2 " 6r cos !
58. x2 # y 2 $ 2ay " 0
57.
x 2 # y 2 $ 2ax " 0
r " 8 sin !
r " 6 cos !
59. y 2 " x3
r " 2a cos !
60.
!r sin !"2 " !r cos !"3
r 2 $ 2a r sin ! " 0
r!r $ 2a sin !" " 0
r"
x2 " y 3
r2 cos2 ! " r3 sin3 !
sin2 ! " r cos3 !
r " 2a sin !
© Houghton Mifflin Company. All rights reserved.
4
1 $ cos !
r"
sin2 !
cos3 !
cos2 !
" cot2 ! csc !
sin3 !
" tan2 ! sec !
61.
r " 6 sin !
r 2 " 6r sin !
x 2 # y 2 " 6y
x 2 # y 2 $ 6y " 0
62.
r " 2 cos !
63.
!"
r2 " 2r cos !
x2 # y 2 " 2x
4%
3
tan ! " tan
$3 "
4% y
"
3
x
y
x
y " $3x
840
Chapter 9
64.
!"
Topics in Analytic Geometry
5%
3
tan ! " tan
65.
5%
" $ $3
3
5%
6
tan ! " tan
y
" $ $3
x
66.
y"
%
, vertical line
2
!"
5% y
"
6
x
11% y
"
6
x
$ $3 y
"
3
x
$ $3
x
3
y"
68. ! " %, horizontal line
69.
$ $3
x
3
r"4
y"0
x"0
11%
6
tan ! " tan
$ $3 y
"
3
x
y # $3x " 0
67. ! "
!"
r 2 " 16
x 2 # y 2 " 16
71.
r " 10
r2 " 100
2
2
x # y " 100
73.
72.
r " $3 csc !
r sin ! " $3
r cos ! " 2
y " $3
x"2
74.
r 2 " cos !
r 2 " sin 2! " 2 sin ! cos !
r 3 " r cos !
r2 " 2
!x 2 # y 2"3#2 " x
% r &% r & "
y
x
2xy
r2
r 4 " 2xy
x 2 # y 2 " x2#3
!x 2 # y 2) 2 " 2xy
!x 2 # y 2"3 " x 2
75.
r " 2 sec !
76.
r " 2 sin 3!
r " 2!3 sin ! $ 4
sin3
r " 3 cos 2!
r " 3!cos2 ! $ sin2 !"
!"
r3 " 3!r2 cos2 ! $ r2 sin2 !"
r 4 " 6r 3 sin ! $ 8r 3 sin3 !
!x2 # y2"3#2 " 3!x2 $ y2" or !x2 # y2"3 " 9!x2 $ y2"2
!x2 # y2"2 " 6!x2 # y2"y $ 8y3
!x2 # y2"2 " 6x2y $ 2y3
77.
r"
1
1 $ cos !
r $ r cos ! " 1
$x 2 # y 2 $ x " 1
x 2 # y 2 " 1 # 2x # x 2
y 2 " 2x # 1
78.
r"
2
1 # sin !
r # r sin ! " 2
$x2
# y2 # y " 2
x2 # y2 " !2 $ y"2
x2 # y2 " 4 $ 4y # y2
x2 # 4y $ 4 " 0
© Houghton Mifflin Company. All rights reserved.
70.
Section 9.6
79.
r"
6
2 $ 3 sin !
80.
r!2 $ 3 sin !" " 6
r"
6
2 cos ! $ 3 sin !
r"
6
2!x#r" $ 3! y#r"
r"
6r
2x $ 3y
1"
6
2x $ 3y
2r " 6 # 3r sin !
2!± $x2 # y2" " 6 # 3y
4!x2 # y2" " !6 # 3y"2
4x2 # 4y2 " 36 # 36y # 9y2
4x2 $ 5y2 $ 36y $ 36 " 0
81.
82.
y
8
r 2 " 49
4
y
10
6
4
2
x 2 # y 2 " 64
2
−8
r"8
r 2 " 64
6
x 2 # y 2 " 49
−4 −2
−2
2
4
6
8
x
Circle of radius 8
centered at origin
−4
−10
!"
tan ! " tan
© Houghton Mifflin Company. All rights reserved.
!"
1
−3
−2
−1
−1
86.
−3
−2
1
2
3
−1
−2
−3
r " 2 csc !
y
4
3
y"2
1
x
−1
1
r sin ! " 2
2
−1
2
Line through origin making
angle of %#6 with positive
x-axis
3
−2
y
3
3y $ $3x " 0
3
2
y
x"3
Vertical line
1
−3
r cos ! " 3
x$3"0
x
−2
r " 3 sec !
7%
6
y
7% $3
" tan ! " tan
"
x
6
3
2
The graph is the line
y " x, which makes an
angle of ! " %#4 with
the positive x-axis.
85.
84.
3
y"x
10
−10
y
%
y
"1"
4
x
2 4 6
−4
−6
−6
%
4
x
−6 −4 −2
−8
83.
841
2x $ 3y " 6
r"7
The graph is a circle
centered at the origin
with radius 7.
Polar Coordinates
1
2
4
−2
−3
87. True, the distances from the origin are the same.
1
y$2"0
Horizontal line through
!0, 2"
−3
−2
−1
x
−1
1
2
3
−2
88. False. For instance when r " 0, any value of !
gives the same point.
x
842
Chapter 9
Topics in Analytic Geometry
89. (a) !r1, !1" " !x1, y1" where x1 " r1 cos !1 and y1 " r1 sin !1.
!r2, !2 " " !x2, y2 " where x2 " r2 cos !2 and y2 " r2 sin !2.
Then x12 # y12 " r12 cos2 !1 # r12 sin2 !1 " r12 and x22 # y22 " r22. Thus,
d " $!x1 $ x2"2 # ! y1 $ y2"2
" $x12 $ 2x1x2 # x22 # y12 $ 2y1y2 # y22
" $!x12 # y12" # !x22 # y22" $ 2!x1x2 # y1y2"
" $r12 # r22 $ 2!r1r2 cos !1 cos !2 # r1r2 sin !1 sin !2"
" $r12 # r22 $ 2r1r2 cos!!1 $ !2".
(b) If !1 " !2, the points are on the same line through the origin. In this case,
(
(
d " $r12 # r22 $ 2r1r2 cos!0" " $!r1 $ r2"2 " r1 $ r2 .
2
2,
(c) If !1 $ !2 " 90', d " $r1 # r2 the Pythagorean Theorem.
% 6 &, %4, 3 & gives d ' 2.053 and%$3, 6 &, %$4, 3 & gives d ' 2.053. (Same!)
(d) For instance, 3,
%
%
7%
4%
90. Answers will vary.
91. cos A "
b2 # c2 $ a2 192 # 252 $ 132
"
" 0.86
2bc
2!19"!25"
A ' 30.7'
cos B "
92. A " 24', a " 10, b " 6
sin B "
a2 # c2 $ b2 132 # 252 $ 192
"
" 0.66615
2ac
2!13"!25"
B ' 48.2'
b sin A
' 0.2440 ⇒ B ' 14.1'
a
C " 180' $ A $ B ' 141.9'
c"
a sin C
' 15.17
sin A
C ' 180' $ 30.7' $ 48.2' ' 101.1'
c
c sin A 12 sin!56'"
a
"
⇒ a"
"
' 16.16
sin A sin C
sin C
sin!38'"
b
c
c sin B 12 sin!86'"
"
"
⇒ b"
' 19.44
sin B sin C
sin C
sin!38'"
95. c2 " a2 # b2 $ 2ab cos C
" 82 # 42 $ 2!8"!4" cos!35'"
' 27.57
94. B " 71', a " 21, c " 29
b2 " a2 # c2 $ 2ac cos B ' 885.458 ⇒ b ' 29.76
sin C " c
sin B
' 0.9214 ⇒ C ' 67.1'
b
A " 180' $ B $ C ' 41.9'
96. B " 64', b " 52, c " 44
sin C "
c sin B
' 0.7605 ⇒ C ' 49.5'
b
A " 180' $ B $ C ' 66.5'
c ' 5.25
b
c
b sin C 4 sin!35'"
"
⇒ sin B "
"
sin B sin C
c
5.25
⇒ B ' 25.9'
A " 180' $ B $ C " 119.1'
a"
b sin A
' 53.06
sin B
© Houghton Mifflin Company. All rights reserved.
93. B " 180' $ 56' $ 38' " 86'
Section 9.6
97. By Cramer’s Rule,
$7
" 5 $ 21 " $16
1
Dx "
$11
$3
Dy "
5 $11
" $15 $ 33 " $48
$3 $3
x"
$7
" $11 $ 21 " $32
1
Dx $32
D y $48
"
" 2, y "
"
"3
D
$16
D
$16
Solution: !2, 3"
D"
( (
( (
( (
3
4
(
(
(
(
(
(
(
(
$10
$5
Dy "
3
4
x"
5
" $20 $ !$25" " 5
$2
10
" $15 $ 40 " $55
$5
Dx
5
D y $55 55
" $ ,, y "
"
"
D
26
D
$26 26
%$ 265 , 55
26 &
100. By Cramer’s Rule,
(
(
(
(
5
D" 1
8
7
$2
$2
(
9
$3 " $89
1
3
D" 2
1
$2
1
$3
1
$3 " 35
9
0
Da " 0
0
$2
1
$3
1
$3 " 0
9
Du "
15
7
0
7
$2
$2
9
$3 " $295
1
3
Db " 2
1
0
0
0
1
$3 " 0
9
5
Dv " 1
8
15
7
0
9
$3 " $844
1
3
Dc " 2
1
$2
1
$3
0
0 "0
0
5
Dw " 1
8
7
$2
$2
15
7 " 672
0
a"
Da
D
D
" 0, b " b " 0, c " c " 0
D
D
D
(
(
(
u"
Du $295 295
D
$844 844
"
"
,v" v"
"
,
D
$89
89
D
$89
89
w"
Dw
672
$672
"
"
D
$89
89
Solution: !0, 0, 0"
© Houghton Mifflin Company. All rights reserved.
5
" $6 $ 20 " $26
$2
Dx "
Solution:
99. By Cramer’s Rule,
843
98. By Cramer’s Rule,
( (
( (
( (
5
D"
$3
Polar Coordinates
Solution:
844 672
,
,$
%295
89 89
89 &
Chapter 9
844
Topics in Analytic Geometry
101. By Cramer’s Rule,
(
(
(
(
$1
D"
2
5
1
3
4
1
Dx " $2
4
1
3
4
$1
Dy "
2
5
1
$2
4
$1
Dz "
2
5
1
3
4
x"
(
(
(
(
2
1 " $15
2
2
1 " $30
2
2
1 " 45
2
1
$2 " $45
4
Dx $30
D
45
"
" 2, y " y "
" $3,
D
$15
D
$15
D
$45
z" z"
"3
D
$15
( (
102. Cramer’s Rule does not apply because
2
D" 2
2
1
2
$1
2
0 " 0.
6
Use elimination to solve the system.
)
2
2
2
1
2
$1
)
)
*
!
!
!
2
0
6
4
5 →
2
2
$R1 # R2 0
$R1 # R3 0
1
1
$2
2
$2
4
$R2 # R1 2
0
2R2 # R3 0
0
1
0
4
$2
0
!
!
!
!
!
!
*
*
4
1 →
$2
3
1
0
Let x3 " a, then x2 " 2a # 1 and x1 " $2a # 32.
Solution: !$2a # 32, 2a # 1, a"
Solution: !2, $3, 3 "
(
4
6
$2
$3
$7
$1
(
1
1 " $20 ( 0
1
The points are not collinear.
105. Points: !$6, $4", !$1, $3", !1.5, $2.5"
(
$6 $4
$1 $3
1.5 $2.5
(
1
1 "0
1
The points are collinear.
104. Points: !$2, 4", !0, 1", !4, $5"
(
$2
0
4
4
1
$5
(
1
1 " 0 ⇒ collinear
1
106. Points: !$2.3, 5", !$0.5, 0", !1.5, $3"
(
$2.3
$0.5
1.5
5
0
$3
(
1
1 " 4.6 ⇒ not collinear
1
© Houghton Mifflin Company. All rights reserved.
103. Points: !4, $3", !6, $7", !$2, $1"
Section 9.7
Section 9.7
■
Graphs of Polar Equations
Graphs of Polar Equations
When graphing polar equations:
1. Test for symmetry
(a) ! " #$2: Replace !r, !" by !r, # % !" or !%r, %!".
(b) Polar axis: Replace !r, !" by !r, %!" or !%r, # % !".
(c) Pole: Replace !r, !" by !r, # $ !" or !%r, !".
(d) r " f !sin !" is symmetric with respect to the line ! " #$2.
(e) r " f !cos !" is symmetric with respect to the polar axis.
2. Find the ! values for which r is maximum.
3. Find the ! values for which r " 0.
##
4. Know the different types of polar graphs.
(a) Limaçons
(b) Rose curves, n ≥ 2
r " a ± b cos !
r " a cos n!
r " a ± b sin !
r " a sin n!
■
(c) Circles
(d) Lemniscates
r " a cos !
r2 " a2 cos 2!
r " a sin !
r2 " a2 sin 2!
r"a
You should be able to graph polar equations of the form r " f !!" with your graphing utility. If your utility
does not have a polar mode, use
x " f !t" cos t
y " f !t" sin t
in parametric mode.
Vocabulary Check
1. ! "
#
2
2. polar axis
3. convex limaçon
5. lemniscate
6. cardioid
1. r " 3 cos 2! is a rose curve.
2. Cardioid
3. Lemniscate
4. r " 3 cos ! is a circle.
5. r " 6 sin 2! is a rose curve.
6. Limaçon
© Houghton Mifflin Company. All rights reserved.
4. circle
7. The graph is symmetric about the line ! " #$2, and
passes through !r, !" " !3, 3#$2". Matches (a).
9. The graph has four leaves. Matches (c).
8. The graph is symmetric about the polar axis and
passes through !r, !" " !3, 0". Matches (c).
10. The graph has three leaves. Matches (d).
845
846
Chapter 9
Topics in Analytic Geometry
11. r " 14 $ 4 cos !
#
!" :
2
12. r " 12 cos 3!
%r " 14 $ 4 cos!% !"
!"
#
:
2
%r " 14 $ 4 cos !
%r " 12 cos 3!
Not an equivalent equation
Not an equivalent equation
r " 14 $ 4 cos!# % !"
r " 12 cos!3!# % !""
r " 14 $ 4!cos # cos ! $ sin # sin !"
r " %12 cos 3!
r " 14 % 4 cos !
Not an equivalent equation
Not an equivalent equation
Polar axis: r " 12 cos!3!% !""
Polar axis: r " 14 $ 4 cos!% !"
r " 12 cos 3!
r " 14 $ 4 cos !
Equivalent equation
Pole:
%r " 12 cos!3!% !""
Equivalent equation
Pole:
%r " 12 cos 3!
%r " 14 $ 4 cos !
Not an equivalent equation
Not an equivalent equation
r " 12 cos!3!# $ !""
r " 14 $ 4 cos!# $ !"
r " %12 cos 3!
r " 14 % 4 cos !
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to polar axis
Answer: Symmetric with respect to polar axis
4
1 $ sin !
#
!" :
2
r"
4
1 $ sin!# % !"
r"
4
1 $ sin # cos ! % cos # sin !
Pole: %r "
4
1 $ sin !
Not an equivalent equation
4
r"
1 $ sin !
r"
4
1 $ sin!# $ !"
Equivalent equation
r"
4
1 % sin !
Polar axis: r "
r"
4
1 $ sin!% !"
4
1 % sin !
Not an equivalent equation
%r "
4
1 $ sin!# % !"
%r "
4
1 $ sin !
Not an equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to ! "
#
2
© Houghton Mifflin Company. All rights reserved.
13. r "
Section 9.7
14. r "
!"
Graphs of Polar Equations
2
1 % cos !
#
:
2
%r "
2
1 % cos!% !"
%r "
2
1 % cos !
Pole: %r "
2
1 % cos !
Not an equivalent equation
Not an equivalent equation
r"
2
1 % cos!# $ !"
r"
2
1 % cos!# % !"
r"
2
1 % !cos # cos ! % sin # sin !"
r"
2
1 % !cos # cos ! $ sin # sin !"
r"
2
1 $ cos !
r"
2
1 $ cos !
Not an equivalent equation
Answer: Symmetric with respect to the polar axis
Not an equivalent equation
Polar axis: r "
r"
2
1 % cos!% !"
2
1 % cos !
Equivalent equation
16. r " 4 csc ! cos ! " 4 cot !
15. r " 6 sin !
#
!" :
2
#
!" :
2
%r " 6 sin!% !"
r " 4 cot !
r " 6 sin !
Equivalent equation
Equivalent equation
Polar axis: %r " 4 cot!# % !"
Polar axis: r " 6 sin!% !"
%r " 4 cot!% !"
r " %6 sin !
r " 4 cot !
© Houghton Mifflin Company. All rights reserved.
Not an equivalent equation
%r " 6 sin!# % !"
Equivalent equation
%r " 6!sin # cos ! % cos # sin !"
Pole:
%r " 4 cot!% !"
Pole:
r " 4 cot!# $ !"
%r " 6 sin !
r " 4 cot !
Not an equivalent equation
Equivalent equation
%r " 6 sin !
Answer: Symmetric with respect to ! "
Not an equivalent equation
polar axis and pole
r " 6 sin!# $ !"
r " %6 sin !
Not an equivalent equation
Answer: Symmetric with respect to ! "
#
2
#
,
2
847
848
Chapter 9
Topics in Analytic Geometry
18. r2 " 25 cos 4!
17. r2 " 16 sin 2!
#
!" :
2
#
!" :
2
!%r"2 " 16 sin!2!% !""
!%r"2 " 25 cos!4!% !""
r2 " 25 cos 4!
r2 " %16 sin 2!
Equivalent equation
Not an equivalent equation
Polar axis: r 2 " 25 cos!4!% !""
r2 " 16 sin!2!# % !""
r2 " 16 sin!2# % 2!"
r2 " 25 cos 4!
r2 " %16 sin 2!
Equivalent equation
!%r"2 " 25 cos 4!
Pole:
Not an equivalent equation
Polar axis: r2 " 16 sin!2!% !""
r2 " 25 cos 4!
r2 " %16 sin 2!
Equivalent equation
Not an equivalent equation
Answer: Symmetric with respect to ! "
!%r" " 16 sin!2!# % !""
2
#
,
2
polar axis and pole
r2 " %16 sin 2!
Not an equivalent equation
Pole:
!%r"2 " 16 sin!2!"
r2 " 16 sin 2!
Equivalent equation
Answer: Symmetric with respect to pole
#
#
20.
#
" 10 1 % sin ! ≤ 10!2" " 20
#1 % sin !# " 2
1 % sin ! " 2
cos ! " 1
or 1 % sin ! " %2
sin ! " %1
!"
3#
2
#r# " #6 $ 12 cos !# ≤ #6# $ #12 cos !#
" 6 $ 12#cos !# ≤ 18
sin ! " 3
or
Not possible
##
Maximum: r " 20 when ! "
r " 0 when 1 % sin ! " 0
sin ! " 1
!"
#
2
3#
2
!"0
##
Maximum: r " 18 when ! " 0
Zero: r " 0 when ! "
2# 4#
,
3 3
© Houghton Mifflin Company. All rights reserved.
## #
19. r " 10!1 % sin !"
Section 9.7
21.
#r# " #4 cos 3!# " 4 #cos 3!# ≤ 4
#cos 3!# " 1
Graphs of Polar Equations
22. r " sin 2!
#r# " #sin 2!#
cos 3! " ± 1
##
Maximum: r " 1 when ! "
# 2#
! " 0, , , #
3 3
# 3# 5# 7#
, , ,
4 4 4 4
#
3#
Zero: r " 0 when ! " 0, , #, , 2#
2
2
# 2#
Maximum: r " 4 when ! " 0, , , #
3 3
##
r " 0 when cos 3! " 0
# # 5#
!" , ,
6 2 6
24. ! " %
23. r " 5
Circle
5#
3
25. r " 3 sin !
Symmetric with respect to
! " #$2
Line
π
2
π
2
Circle with radius of 3$2
π
2
2
4
6
8
0
1
0
2
1
27. r " 3!1 % cos !"
26. r " 2 cos !
Circle
3
2
0
28. r " 4!1 $ sin !"
Cardioid
Cardioid
Radius: 1, center: !1, 0"
π
2
π
2
© Houghton Mifflin Company. All rights reserved.
π
2
2
1
3
4
0
4
0
30. r " 1 % 2 sin !
29. r " 3 % 4 cos !
π
2
π
2
6
8
0
Limaçon
π
2
1
1 2
8
31. r " 4 $ 5 sin !
Limaçon with inner loop
Limaçon
6
2
0
0
4
0
849
Chapter 9
850
Topics in Analytic Geometry
32. r " 3 $ 6 cos !
33. r " 5 cos 3!
Limaçon
Rose curve
Rose curve
π
2
π
2
π
2
4 6 8 10 12
34. r " %sin 5!
0
0
5 6
1
37. r " 8 cos 2!
36. r " 3 cos 5!
35. r " 7 sin 2!
Rose curve, five petals
Rose curve, four petals
π
2
0
1
12
π
2
−18
18
−12
4 5 6
38.
1
2
3
4
39. r " 2!5 % sin !"
2
40.
4
−12
10
−3
0 ≤ ! < 2#
0
0
12
3
−18
18
−12
−2
0 ≤ ! ≤ 2#
0 ≤ ! ≤ 2#
−14
41. r " 3 % 6 cos !, 0 ≤ ! ≤ 2#
42.
−6
6
43. r "
2
3
#
, 0≤!≤
sin ! % 2 cos !
2
6
4
−14
4
−6
−6
0 ≤ ! ≤ 2#
−6
44.
−4
45. r 2 " 4 cos 2!, %2# ≤ ! ≤ 2#
4
6
−3
3
−4
0 ≤ ! ≤ 2#
46. r 2 " 9 sin !
r " ± 3%sin !
2
−6
6
Graph both functions using
0 ≤ ! ≤ 2#.
4
−2
−6
6
−4
© Houghton Mifflin Company. All rights reserved.
0 ≤ ! < 2#
Section 9.7
47. r " 4 sin ! cos2 !, 0 ≤ ! ≤ #
48.
Graphs of Polar Equations
851
49. r " 2 csc ! $ 6
2
2
14
−3
−3
3
3
−18
−2
18
0 ≤ ! ≤ #
−2
−10
0 ≤ ! < 2#
50.
52. r " e!$2
51. r " e2!
12
200
−18
18
15
−2000
400
−15
30
−12
0 ≤ ! ≤ 2#
−1400
−15
Answers will vary.
53. r " 3 % 2 cos !, 0 ≤ ! < 2#
54.
Answers will vary.
55. r " 2 cos
1
−6
6
& 2 ', 0 ≤ ! < 4#
3!
4
2
−6
6
−3
−7
0 ≤ ! < 2#
−4
56.
−2
57. r2 " sin 2!, 0 ≤ ! <
4
−6
#
2
!Use r1 " %sin 2! and r2 " % %sin 2!."
6
1
−4
−1
© Houghton Mifflin Company. All rights reserved.
0 ≤ ! < 4#
1
−1
58.
59. r " 2 % sec !
1
−1.5
1.5
0 < ! <
r"
−1
3
&
x " %1 is an asymptote.
±1
%!
4
−6
6
−4
Chapter 9
852
Topics in Analytic Geometry
60.
r " 2 $ csc ! " 2 $
1
sin !
5
r sin ! " 2 sin ! $ 1
−6
6
r!r sin !" " 2r sin ! $ r
!
"
−3
!
"
± %x 2 $ y 2 ! y" " 2y $ ± %x 2 $ y 2
!± %x
2
$ y " ! y % 1" " 2y
2
2y
!± %x 2 $ y 2" " y % 1
x2 $ y2 "
x2 "
4y 2
! y % 1" 2
y 2!3 $ 2y % y 2"
! y % 1" 2
x"±
%y !3! $y %2y1%" y "
2
2
"±
2
The graph has an asymptote at y " 1.
61. r "
2
!
# #
y
%3 $ 2y % y 2
y%1
62.
3
y " 2 is an asymptote.
4
−6
−3
6
3
−1
−4
64. False. For example, let r " cos 3!.
63. True. It has five petals.
65. r " cos!5!" $ n cos !, 0 ≤ ! < #; Answers will vary.
−6
4
6
−6
−4
6
2
−3
−2
3
−6
6
3
−3
6
n"2
−6
6
−4
n"5
3
−2
4
−4
n"4
2
n"1
−6
−4
n " %2
−3
4
3
−2
−2
n"0
4
−3
2
−2
n " %1
6
n " %3
2
3
2
−4
n " %4
−3
−6
−4
n " %5
n"3
4
© Houghton Mifflin Company. All rights reserved.
4
Section 9.7
Graphs of Polar Equations
66. The graph of r " f !!" is rotated about the pole through an angle '. Let !r, !" be any point
on the graph of r " f !!". Then !r, ! $ '" is rotated through the angle ', and since
r " f !!! $ '" % '" " f !!", it follows that !r, ! $ '" is on the graph of r " f !! % '".
67. Use the result of Exercise 66.
&
68. (a) r " 2 % sin ! %
#
(a) Rotation: ' "
2
"2%
Original graph: r " f !sin !"
& &
Rotated graph: r " f sin ! %
#
2
'' " f !%cos !"
Rotated graph: r " f !sin!! % #"" " f !%sin !"
& &
Rotated graph: r " f sin ! %
&
'+
#
6
#
3
*&
3#
2
#
2
70. (a) r " 1 % sin !
y
x
–2
1
'+
–3
© Houghton Mifflin Company. All rights reserved.
&
(b) r " 1 % sin ! %
#
4
'
y
*&
2#
(c) r " 2 sin 2 ! %
3
'+
'
1
x
–2
–1
1
–2
" %3 cos 2! % sin 2!
(d) r " 2 sin(2!! % #")
" 2 sin!2! % 2#"
" 4 sin ! cos !
2
–1
" %4 sin ! cos !
" 2 sin 2!
'
1
" %2 sin 2!
4#
" 2 sin 2! %
3
3#
2
'' " f !cos !"
" 2 sin!2! % #"
&
'
r " 2 % cos !
'
" sin 2! % %3 cos 2!
(b) r " 2 sin 2 ! %
#
2
r " 2 $ sin !
(d) r " 2 % sin ! %
Original graph: r " f !sin !"
" 2 sin 2! %
&
(b) r " 2 % sin ! %
&
3#
2
*&
!sin ! % cos !"
(c) r " 2 % sin!! % #"
Original graph: r " f !sin !"
69. (a) r " 2 sin 2 ! %
2
'
r " 2 $ cos !
(b) Rotation: ' " #
(c) Rotation: ' "
%2
#
4
–3
2
853
Chapter 9
854
Topics in Analytic Geometry
71. r ! 2 $ k cos #
k!0
k!1
k!2
k!3
Circle
Convex limaçon
Cardioid
Limaçon with inner loop
4
−6
6
−6
−4
6
−4
4
4
4
8
−4
−4
−4
−4
8
72. r ! 3 sin k #
(a)
(b)
4
−6
−6
6
k ! 2.5: 0 ≤ # < 4%
k ! 1.5: 0 ≤ # < 4%
■
6
−4
−4
Section 9.8
(c) Yes. Answers will vary.
4
Polar Equations of Conics
The graph of a polar equation of the form
r!
ep
ep
or r !
1 ± e cos #
1 ± e sin #
!!
is a conic, where e > 0 is the eccentricity and p is the distance between the focus (pole) and the directrix.
(a) If e < 1, the graph is an ellipse.
(b) If e ! 1, the graph is a parabola.
(c) If e > 1, the graph is a hyperbola.
■
Guidelines for finding polar equations of conics:
ep
1 $ e sin #
ep
(b) Horizontal directrix below the pole: r !
1 " e sin #
(c) Vertical directrix to the right of the pole: r !
(d) Vertical directrix to the left of the pole: r !
© Houghton Mifflin Company. All rights reserved.
(a) Horizontal directrix above the pole: r !
ep
1 $ e cos #
ep
1 " e cos #
Vocabulary Check
1. conic
2. eccentricity, e
3. (a) i
(b) iii
(c) ii
Section 9.8
1. r !
2e
1 $ e cos #
(a) Parabola
2. (a) Parabola
4
a
−4
c
b
8
b
−8
2e
1 " e sin #
4. (a) Parabola
4
b a
−9
(c) Hyperbola
8
c
(b) Ellipse
9
−9
(c) Hyperbola
c
9
b
a
−4
−8
4
1 " cos #
6. r !
e ! 1 ⇒ parabola
e!
Vertical directrix to left of pole
3
3"2
!
2 " cos # 1 " #1"2$ cos #
1
⇒ ellipse
2
Vertical directrix to left of pole
Matches (b).
Matches (c).
7. r !
8. r !
4
1 " 3 sin #
9. r !
3
1 $ 2 sin #
e ! 2 ⇒ hyperbola
Vertical directrix to right of pole
Horizontal directrix below
the pole.
Horizontal directrix above
the pole.
Matches (f).
Matches (e).
Matches (d).
10. r !
© Houghton Mifflin Company. All rights reserved.
3
3"2
!
2 $ cos # 1 $ #1"2$ cos #
e ! 3 ⇒ hyperbola
e!
1
⇒ ellipse
2
4
1 $ sin #
e ! 1 ⇒ parabola
Vertex:
%2, %2 &
11. r !
2
1 " cos #
12. r !
4
1
!
4 " cos # 1 " #1"4$ cos #
e ! 1 ⇒ parabola
Vertex: #r, #$ ! #1, %$
Vertex: #1, %"2$
π
2
1
e ! , p ! 4, ellipse
4
1
Vertices:
#r, #$ !
%43, 0&, %45, %&
2
1 $ sin #
e ! 1 ⇒ parabola
Matches (a).
13. r !
4
−4
−4
(b) Ellipse
5. r !
a
(c) Hyperbola
(c) Hyperbola
(a) Parabola
855
4
c
(b) Ellipse
(b) Ellipse
3. r !
Polar Equations of Conics
2
0
Chapter 9
14. r !
e!
Topics in Analytic Geometry
7
1
!
7 $ sin # 1 $ #1"7$ sin #
1
⇒ ellipse
7
Vertices: #r, #$ !
16. r !
e!
e!
%78, %2 &, %76, 32%&
17. r !
e!
1
6
6
#1"2$#6$
!
2 $ sin # 1 $ #1"2$ sin #
1
⇒ ellipse
2
% &%
5
"5
!
"1 $ 2 cos # 1 " 2 cos #
π
2
19. r !
&
1 2
e ! 2 ⇒ hyperbola
&
π
2
Vertices:
10
10"3
!
3 $ 9 sin # 1 $ 3 sin #
21. r !
e ! 3 ⇒ hyperbola
"5
1 " sin #
&% &
22. r !
−6
−3
6
1
1 $ #17"14$ sin #
Hyperbola
24. r !
9
12
2 " cos #
Ellipse
−9
3
−2
−4
14
14 $ 17 sin #
"1
2 $ 4 sin #
2
4
%56, %2 &, %" 53, 32%&
1 2 3 4 5
Hyperbola
Parabola
Vertices:
!
0
Hyperbola
%
23. r !
4 5
3
3"4
!
4 " 8 cos # 1 " 2 cos #
3
1
#r, #$ ! " , 0 , , %
4
4
#r, #$ !
0
%87, %2 &, %8, 32%&
%
3%
2, , 6,
2
2
e ! 2 ⇒ hyperbola
20. r !
3 4 5 6
Vertices:
% 5, %&
%
π
2
3
⇒ ellipse
4
#r, #$ !
2
⇒ ellipse
3
5
Vertices: #5, 0$, " , %
3
8
2
!
4 $ 3 sin # 1 $ #3"4$ sin #
Vertices:
6
2
!
3 " 2 cos # 1 " #2"3$ cos #
Vertices: #6, 0$,
18. r !
15. r !
10
−15
15
9
−3
−10
0
© Houghton Mifflin Company. All rights reserved.
856
Section 9.8
25.
26.
2
−4
2
27.
2
−5
Polar Equations of Conics
3!
1
−2!
−2
−!
−2
Hyperbola
Ellipse
28.
4!
2
29.
−3
30.
2
3
−9
3
−3
9
3
−2
−2
31.
−9
32.
4
1
−8
−9
−7
−8
33. e ! 1, x ! "1, p ! 1
Vertical directrix to the left of the pole
r!
1
1#1$
!
1 " 1 cos # 1 " cos #
1
35. e ! , y ! 1, p ! 1
2
Horizontal directrix above the pole
© Houghton Mifflin Company. All rights reserved.
r!
4
9
#1"2$#1$
1
!
1 $ #1"2$ sin # 2 $ sin #
37. e ! 2, x ! 1, p ! 1
Vertical directrix to the right of the pole
2#1$
2
r!
!
1 $ 2 cos # 1 $ 2 cos #
39. Vertex:
%1, " 2 & ⇒ e ! 1, p ! 2
%
Horizontal directrix below the pole
1#2$
2
r!
!
1 " 1 sin # 1 " sin #
34. e ! 1, y ! "4, p ! 4
Horizontal directrix below the pole
r!
1#4$
4
!
1 " #1$ sin # 1 " sin #
3
36. e ! , y ! "4, p ! 4
4
Horizontal directrix below pole
r!
(3"4$4
12
!
4
"
3 sin #
1 " #3"4$ sin #
3
38. e ! , x ! "1, p ! 1
2
Vertical directrix to the left of the pole
r!
3"2#1$
3
!
1 " #3"2$ cos # 2 " 3 cos #
40. Parabola, e ! 1, vertex: #8, 0$
Vertical directrix to right of pole
r!
ep
16
!
1 $ e cos # 1 $ cos #
857
Chapter 9
Topics in Analytic Geometry
41. Vertex: #5, %$ ⇒ e ! 1, p ! 10
42. Vertex:
Vertical directrix to left of pole
r!
2
3
Vertical directrix to the right of the pole
44. Center: 1,
10
3 $ 2 cos #
r!
#1&3$ p
p
!
1 $ #1&3$ sin # 3 $ sin #
2!
p
3 $ sin#%"2$
p!8
r!
45. Center: #8, 0$, c ! 8, a ! 12, e !
c
2
!
a 3
20 !
r!
%5, 2 &, c ! 5, a ! 4, e ! 4
3%
r!
#5"4$p
5p
!
1 " #5"4$ sin # 4 " 5 sin #
2p
! 2p ⇒ p ! 10
3"2
1!
5p
4 " 5 sin#3%"2$
20
3 " 2 cos #
p!
9
5
r!
5#9"5$
9
!
4 " 5 sin # 4 " 5 sin #
%52, %2 &, c ! 52, a ! 32, e ! 53
48. Center:
Horizontal directrix above the pole
%
&
%
"3%
3%
rather than "1,
2
2
in order to get a directrix between the vertices.
1!
5p
3 $ 5 sin#"3%"2$
p!
8
5
r!
5#8"5$
8
!
3 $ 5 sin # 3 $ 5 sin #
%52, %2 &, c ! 52, a ! 32, e ! ac ! 53
Horizontal directrix above the pole
#5"3$p
5p
!
1 $ #5"3$ sin # 3 $ 5 sin #
Substitute the point 1,
5
Horizontal directrix below the pole
#2"3$p
2p
!
1 " #2"3$ cos # 3 " 2 cos #
47. Center:
r!
8
3 $ sin #
46. Center:
Vertical directrix to left of pole
r!
&
3%
1
, c ! 1, a ! 3, e !
2
3
Horizontal directrix above the pole
#2"3$p
2p
r!
!
1 $ #2"3$ cos # 3 $ 2 cos #
r!
1#20$
20
!
1 $ 1 sin # 1 $ sin #
%
43. Center: #4, %$, c ! 4, a ! 6, e !
2p
2p
!
⇒ p!5
3 $ 2 cos 0
5
%
Horizontal directrix above pole
1#10$
10
r!
!
1 " 1 cos # 1 " cos #
2!
%10, 2 & ⇒ e ! 1, p ! 20
&
r!
#5"3$p
5p
!
1 $ #5"3$ sin # 3 $ 5 sin #
1!
5p
8
⇒ p!
3 $ 5 sin#%"2$
5
r!
8
3 $ 5 sin #
© Houghton Mifflin Company. All rights reserved.
858
Section 9.8
49. When # ! 0, r ! c $ a ! ea $ a ! a#1 $ e$.
r!
ep
1 " e cos 0
r!
a#1 " e2$ ! ep.
51. r !
(1 " #0.0167$2)#92.956 ' 106$
1 " 0.0167 cos #
52. a ! 35.983
r!
'
Perihelion distance:
r ! 92.956
'
106#1 " 0.0167$ ' 9.1404
'
107
'
r!
'
Perihelion:
107#1 " 0.0484$ ' 7.4073
'
108 km
#1 $ 0.0484$ ' 8.1609 '
107
55. a ! 4.498
'
109, e ! 0.0086, Neptune
a ! 5.906
'
109, e ! 0.2488, Pluto
Pluto: r !
'
'
'
107 miles
107 miles
107, e ! 0.0542
#1 " 0.05422$#142.673 ' 107$
1 " 0.0542 cos #
1.4225 ' 109
1 " 0.0542 cos #
Perihelion distance: a#1 " e$ ' 1.3494
Aphelion:
(a) Neptune: r !
3.4462 ' 107
1 " 0.2056 cos #
54. a ! 142.673
7.7659 ' 108
!
1 " 0.0484 cos #
r ! 77.841 '
#1 " 0.20562$#35.983 ' 106$
1 " 0.2056 cos #
107
#1 " 0.04842$77.841 ' 107
1 " 0.0484 cos #
'
106, e ! 0.2056
Aphelion distance: a#1 $ e$ ' 4.3381
r ! 92.956 ' 106#1 $ 0.0167$ ' 9.4508
r ! 77.841
'
Perihelion distance: a#1 " e$ ' 2.8585
Aphelion distance:
© Houghton Mifflin Company. All rights reserved.
#1 " e2$a
#1 " e$#1 $ e$a
!
! a#1 $ e$
1 " e cos 0
1"e
ep
#1 " e2$a
!
.
1 " e cos # 1 " e cos #
9.2930 ' 107
'
1 " 0.0167 cos #
53. r !
#1 " e2$a
#1 " e$#1 $ e$a
!
! a#1 " e$
1 " e cos %
1$e
Maximum distance occurs when # ! 0.
a#1 $ e$#1 " e$ ! ep
Thus, r !
859
50. Minimum distance occurs when # ! %.
Therefore,
a#1 $ e$ !
Polar Equations of Conics
Aphelion distance: a#1 $ e$ ' 1.5041
108
'
'
109 km
109 km
km
#1 " 0.0086 2$4.498 ' 109
4.4977 ' 109
!
1 " 0.0086 cos #
1 " 0.0086 cos #
#1 " 0.24882$5.906 ' 109
5.5404 ' 109
!
1 " 0.2488 cos #
1 " 0.2488 cos #
(b) Neptune: Perihelion: 4.498
Aphelion: 4.498
Pluto: Perihelion: 5.906
'
'
'
109#1 " 0.0086$ ' 4.4593
109#1 $ 0.0086$ ' 4.5367
109#1 " 0.2488$ ' 4.4366
Aphelion: 5.906 ' 109#1 $ 0.2488$ ' 7.3754
'
'
'
'
109 km
(c)
7 × 109
109 km
109 km
−5 × 109
109 km
(d) Yes. Pluto is closer to the sun for just a very short time. Pluto was considered the
ninth planet because its mean distance from the sun is larger than that of Neptune.
(e) Although the graphs intersect, the orbits do not, and the planets won’t collide.
8 × 109
−7 × 109
860
Chapter 9
Topics in Analytic Geometry
56. (a) Radius of earth ' 4000 miles. Choose r !
ep
.
1 " e cos #
Vertices: #126,800, 0$ and #4119, %$
a!
126,800 $ 4119
! 65,459.5
2
c ! 65,459.5 " 4119 ! 61,340.5
e!
c
61,340.5
!
' 0.937
a 65,459.5
2a !
ep
ep
ep
ep
2ep
$
!
$
!
1 " e cos 0 1 " e cos#%$ 1 " e 1 $ e 1 " e2
Thus, p !
a#1 " e2$
ep
7988.1
' 8525.2. Thus, r !
'
.
e
1 " e cos # 1 " 0.937 cos #
(b) When # ! 60(, r ' 15,029 and the distance from the surface of the earth to the satellite is
15,029 " 4000 ! 11,029 miles.
(c) When # ! 30(, r ' 42,370 and distance ! 38,370 miles.
4
"4"3
!
"3 " 3 sin # 1 $ sin #
57. r !
58. r2 !
16
%
9 " 4 cos # $
False. The directrix is below the pole.
%
4
&
False. The graph is not an ellipse.
(It is two ellipses.)
x2
y2
"
!1
a2 b2
60.
r 2 cos2 # r 2 sin2 #
$
!1
a2
b2
r2 cos2 # r2 sin2 #
"
!1
a2
b2
r 2 cos2 # r 2#1 " cos2 #$
$
!1
a2
b2
r2 cos2 # r2#1 " cos2 #$
"
!1
a2
b2
r 2b 2 cos2 # $ r 2a2 " r 2a2 cos2 # ! a 2 b 2
r2b2 cos2 # " r2a2 $ r2a2 cos2 # ! a2b2
r 2#b 2 " a2$ cos2 # $ r 2a2 ! a 2 b 2
r2#b2 $ a2$ cos2 # " r2a2 ! a2b2
a2 $ b2 ! c2
For an ellipse, b2 " a2 ! "c2. Hence,
r2c2 cos2 # " r2a2 ! a2b2
"r 2c 2 cos2 # $ r 2a2 ! a 2 b 2
"r 2
%&
c
a
2
cos2 # $ r 2 ! b2, e !
c
a
"r 2e 2 cos2 # $ r 2 ! b2
r 2#1 " e 2 cos2 #$ ! b2
b2
.
r2 !
1 " e 2 cos2 #
r2
%ac &
2
cos2 # " r2 ! b2, e !
c
a
r2e2 cos2 # " r2 ! b2
r2#e2 cos2 # " 1$ ! b2
r2 !
b2
e2 cos2 # " 1
!
"b2
1 " e2 cos2 #
© Houghton Mifflin Company. All rights reserved.
x2 y 2
$ 2!1
a2
b
59.
Section 9.8
61.
x2
y2
$
!1
169 144
62.
a ! 13, b ! 12, c ! 5, e !
r2 !
63.
x2
y2
"
!1
9
16
a ! 3, b ! 4, c ! 5, e !
144
24,336
!
2
1 " #25"169$ cos # 169 " 25 cos2 #
r2 !
5
3
"16
144
!
1 " #25"9$ cos2 # 25 cos2 # " 9
x2
y2
$
!1
25 16
a ! 5, b ! 4, c ! 3, e !
r2 !
64.
5
13
Polar Equations of Conics
3
5
b2
16
400
!
!
1 " e2 cos2 # 1 " #9"25$ cos2 # 25 " 9 cos2 #
x2
y2
" !1
36
4
a ! 6, b ! 2, c ! *40 ! 2 *10, e !
r2 !
2*10 *10
!
6
3
"b2
"4
36
!
!
1 " e2 cos2 # 1 " #10"9$ cos2 # 10 cos2 # " 9
65. Center: #x, y$ ! #0, 0$, c ! 5, a ! 4, e !
5
4
b2 ! c2 " a2 ! 25 " 16 ! 9 ⇒ b ! 3
r2 !
"b2
"9
144
!
!
2
2
2
1 " e cos # 1 " #25"16$ cos # 25 cos2 # " 16
© Houghton Mifflin Company. All rights reserved.
66. Center: #x, y$ ! #0, 0$, c ! 4, a ! 5, e !
4
5
b2 ! a2 " c2 ! 25 " 16 ! 9 ⇒ b ! 3
r2 !
67. r !
b2
9
225
!
!
2
2
2
1 " e cos # 1 " #16"25$ cos # 25 " 16 cos2 #
4
1 " 0.4 cos #
Vertical directrix to left of pole
(b) r !
(a) e ! 0.4 ⇒ ellipse
6
4
1 $ 0.4 cos #
Vertical directrix to right of pole
7
Graph is reflected in line # ! %"2.
−9
9
−9
9
r!
−6
−5
4
1 " 0.4 sin #
Horizontal directrix below pole
90( rotation counterclockwise
861
862
Chapter 9
Topics in Analytic Geometry
68. The lengths of the major and
minor axes increase as p
increases.
70.
69. Answers will vary.
r 2 ! a r sin # $ b r cos #
x2 $ y2 ! ay $ bx
#0.5$2
Example: r !
1 $ #0.5$ sin #
r!
r ! a sin # $ b cos #
Circle
#0.5$4
1 $ #0.5$ sin #
72. 6 cos x " 2 ! 1
71. 4*3 tan # " 3 ! 1
tan # !
#!
*3
1
!
3
*3
cos x !
%
$ n%
6
x!
73. 12 sin2 # ! 9
1
2
sin2 # !
%
5%
$ 2n%,
$ 2n%
3
3
sin # ! ±
#!
74. 9 csc2 x " 10 ! 2
csc2 x !
4
3
sin2 x !
3
4
sin x !
x!
75. 2 cot x ! 5 cos
%
2
2
%
2%
$ n%,
$ n%
3
3
%
4
*2 sec # ! 2*2
cot x ! 0
± *3
*3
76. *2 sec # ! 2 csc
2 cot x ! 0
x!
3
4
sec # ! 2
%
$ n%
2
cos # !
2
#!
%
2%
$ n% ,
$ n%
3
3
1
2
%
5%
$ 2n%,
$ 2n%
3
3
77. cos#u $ v$ ! cos u cos v " sin u sin v
!
!
78. sin#u $ v$ ! sin u cos v $ sin v cos u
% & % &%" *12&
4 1
3
" "
5 *2
5
1
5*2
!
*2
10
79. sin#u " v$ ! sin u cos v " sin v cos u
% 35&%*12& " %" *12&%45&
81.
12C9
! 220
1
5*2
!
*2
10
82.
18C16
! 153
%"35&%*12& $ %*"12&%45&
!
"7
"7*2
!
5*2
10
80. cos#u " v$ ! cos u cos v $ sin u sin v
! "
!
!
83.
10 P3
!
%45&%*12& $ %"35&%*"12&
!
7
7*2
!
5*2
10
! 720
84.
29P2
! 812
© Houghton Mifflin Company. All rights reserved.
4
1
1
3
For Exercises 77–80: sin u ! " , cos u ! , cos v !
, sin v ! "
5
5
*2
*2
Review Exercises for Chapter 9
Review Exercises for Chapter 9
1. Radius " !"#3 # 0#2 ! "#4 # 0#2
2. Radius " !"8 # 0#2 ! "#15 # 0#2
" !9 ! 16 " !25 " 5
" !64 ! 225 " !289 " 17
x 2 ! y 2 " 25
x 2 ! y 2 " 289
1
3. Radius " !"5 # "#1##2 ! "6 # 2#2
2
1
4. Radius " !"6 # "#2##2 ! "#5 # 3#2
2
1
1
" !36 ! 16 " !52 " !13
2
2
Center "
1
" !64 ! 64 " 4!2
2
$5 ! 2"#1#, 6 !2 2% " "2, 4#
Center "
"x # 2#2 ! " y # 4#2 " 13
5.
"x # 2#2 ! " y ! 1#2 " 32
1 2 1 2
x ! y " 18
2
2
6.
x 2 ! y 2 " 36
3 2 3 2
x ! y "1
4
4
x2 ! y2 "
Center: "0, 0#
Radius:
16x 2 ! 16y 2 # 16x ! 24y # 3 " 0
9
16"x 2 # x ! 14 # ! 16" y 2 ! 32 y ! 16
#"3!4!9
16"x # 12 # ! 16" y ! 34 # " 16
2
2
"x # 12 #2 ! " y ! 34 #2 " 1
© Houghton Mifflin Company. All rights reserved.
Center:
"12, # 34 #
Radius: 1
8.
4x 2 ! 4y 2 ! 32x # 24y ! 51 " 0
4"x 2 ! 8x ! 16# ! 4" y 2 # 6y ! 9# " #51 ! 64 ! 36
4"x ! 4#2 ! 4" y # 3#2 " 49
"x ! 4#2 ! " y # 3#2 " 494
Center: "#4, 3#
Radius:
7
2
4
3
Center: "0, 0#
Radius: 6
7.
$#22! 6, 3 #2 5% " "2, #1#
2
!3
"
2!3
3
863
Chapter 9
Topics In Analytic Geometry
9. "x 2 ! 4x ! 4# ! " y 2 ! 6y ! 9# " 3 ! 4 ! 9
10. "x 2 ! 8x ! 16# ! " y 2 # 10y ! 25# " 8 ! 16 ! 25
"x ! 2#2 ! " y ! 3#2 " 16
"x ! 4#2 ! " y # 5#2 " 49
Center: "#2, #3#
Center: "#4, 5#
y
2
Radius: 4
−7 −6
−4 −3 −2 −1
2 3
x
y
14
Radius: 7
10
8
6
4
2
−2
−3
−4
−5
−6
−14
−10
2 4 6
−4
−6
−8
12. x-intercepts: "x ! 5#2 ! "0 # 6#2 " 27
11. x-intercepts: "x # 3#2 ! "0 ! 1#2 " 7
"x ! 5#2 " #9, impossible
"x # 3#2 " 6
No x-intercepts
x # 3 " ± !6
y-intercepts: "0 ! 5#2 ! " y # 6#2 " 27
x " 3 ± !6
"3 ± !6, 0#
" y # 6#2 " 2
y-intercepts: "0 # 3#2 ! " y ! 1#2 " 7
y # 6 " ± !2
" y ! 1#2 " #2, impossible
y " 6 ± !2
"0, 6 ± !2 #
No y-intercepts
13. 4x # y 2 " 0
4
x
–2
2
4
6
8
10
x
–12 –9 –6
Vertex: "0, 0#
2
–2
y
3
x 2 " 4"#2#y, p " #2
6
Vertex: "0, 0#
Directrix: x " #1
14. y " # 18 x 2
y
y 2 " 4"1#x, p " 1
Focus: "1, 0#
6
–9
Focus: "0, #2#
–12
–15
Directrix: y " 2
–4
9 12
–6
–18
–21
–6
16. 14 y # 8x 2 " 0
1
15. 2 y 2 ! 18x " 0
1 2
2y
x
8x 2 " 14 y
" #18x
1
1
x 2 " 32
y " 4"128
#y, p " 1281
y 2 " #36x " 4"#9#x, p " #9
Vertex: "0, 0#
Vertex: "0, 0#
y
1
Focus: "0, 128
#
12
Focus: "#9, 0#
Directrix: x " 9
Directrix: y "
4
−20 −16 −12 −8
−4
y
5
16
1
4
1
# 128
3
16
1
8
x
−4
4
−3
16
−1 − 1
8
16
)0, 1281 )
(0, 0)
x
1
8
3
16
© Houghton Mifflin Company. All rights reserved.
864
Review Exercises for Chapter 9
18. Vertex: "4, 2#
17. Vertex: "0, 0#
Focus: "#6, 0#
Focus: "4, 0#
Parabola opens to left.
Vertical axis, p " #2
y 2 " 4px
"x # 4#2 " 4"#2#" y # 2#
y 2 " 4"#6#x
"x # 4#2 " #8" y # 2#
y 2 " #24x
20. Vertex: "0, 5#
19. Vertex: "#6, 4#
Passes through "0, 0#
" y # 5#2 " 4p"x # 0# " 4px
Vertical axis
"6, 0# on graph:
"x ! 6# " 4p" y # 4#
"0 # 5#2 " 4p"6# ⇒ p " 25
24
"0 ! 6#2 " 4p"0 # 4#
25
" y # 5#2 " 4" 25
24 #x " 6 x
2
36 " #16p
# 94 " p
"x ! 6#2 " 4"# 94 #" y # 4#
"x ! 6#2 " #9" y # 4#
$ 12%y, p " # 12
22.
21. x2 " #2y " 4 #
Focus:
1
!b
2
d2 "
!
$
"2 # 0#2 ! #2 !
1
2
%
2
"
5
2
1
5
!b" ⇒ b"2
2
2
b!2
4
Slope of tangent line:
"
" #2
0 # 2 #2
Equation: y ! 2 " #2"x # 2#
y " #2x ! 2
x-intercept: "1, 0#
(0, b)
1
2
$# 12, 0%
Let "b, 0# be the x-intercept of the tangent line.
d1 "
1
!b
2
d2 "
!$#8 ! 12%
17
1
!b"
⇒ b"8
2
2
#4 # 0 1
"
#8 # 8 4
1
y # 0 " "x # 8#
4
F
−2
2
x
−1
d2
−2
Focus:
m"
y
d1
%
p"#
d1 " d2 ⇒
© Houghton Mifflin Company. All rights reserved.
$
1
4 # x " y2
2
$0, # 12%
d1 "
#2x " y 2
(2, − 2)
1
y" x#2
4
x-intercept: "8, 0#
2
! "#4 # 0#2 "
17
2
865
866
Chapter 9
Topics In Analytic Geometry
23. x2 " 4p" y # 12#
"4, 10# on curve:
16 " 4p"10 # 12# " #8p ⇒ p " #2
x 2 " 4"#2#" y # 12# " #8y ! 96
y"
#x2 ! 96
8
y " 0 if x 2 " 96 ⇒ x " 4!6 ⇒ width is 8!6 meters.
24. (a) Parabola:
(b) Parabola:
x2 " #4" y # 4# ⇒ y " # 14 x2 ! 4
Vertex: "0, 4#
Passes through "± 4, 0#
x2
Circle:
x2 ! "y ! 4!3# " 64 ⇒ y " !64 # x2 # 4!3
2
" 4p" y # 4#
16 " 4p"0 # 4#
d " "# 14 x2 ! 4# # "!64 # x2 # 4!3 #
16 " #16p
d " # 14 x2 # !64 # x2 ! 4 ! 4!3
#1 " p
x2 " #4" y # 4#
Circle:
x
0
1
2
3
4
d
2.928
2.741
2.182
1.262
0
Passes through "± 4, 0#
Radius: r " 8
Center: "0, k#
x2 ! " y # k#2 " 82
"± 4#2 ! "0 # k#2 " 82
16 ! k2 " 64
k2 " 48
k " # !48 " #4!3
x2 ! " y ! 4!3 # " 64
25.
x2
y2
!
"1
4
16
26.
x2 y 2
! "1
9
8
a " 4, b " 2, c " !16 # 4 " !12 " 2!3
a " 3, b " 2!2, c " !9 # 8 " 1
Center: "0, 0#
Center: "0, 0#
Vertices: "0, ± 4#
Foci: "0, ± 2!3 #
Eccentricity "
"
"
c
a
2!3
4
!3
2
Vertices: "± 3, 0#
y
Foci: "± 1, 0#
3
2
1
−4 −3
−1
–2
y
4
1
3
4
x
Eccentricity "
c
1
"
a 3
2
1
−4
−2 −1
−1
−2
–3
−4
x
1
2
4
© Houghton Mifflin Company. All rights reserved.
2
Review Exercises for Chapter 9
27.
"x # 4#2 " y ! 4#2
!
"1
6
9
28.
"x ! 1#2 " y # 3#2
!
"1
16
6
a " 3, b " !6, c " !9 # 6 " !3
a " 4, b " !6, c " !16 # 6 " !10
Center: "4, #4#
Center: "#1, 3#
y
Vertices: "4, #1#, "4, #7#
Foci: "4, #4 ± !3 #
Eccentricity "
1
−2 −1
!3
c
"
a
3
1 2 3
5 6 7 8
x
−2
−3
−4
−5
−6
−7
−8
−9
y
9
8
7
6
5
4
Vertices: "3, 3#, "#5, 3#
Foci: "#1 ± !10, 3 #
Eccentricity "
29. (a) 16"x2 # 2x ! 1# ! 9" y2 ! 8y ! 16# " #16 ! 16 ! 144
!10
c
"
a
4
2
1
−6 −5 −4 −3 −2 −1
(c)
16"x # 1#2 ! 9" y ! 4#2 " 144
y
−3 −2 −1
1
2
3
4
5
x
−2
"x # 1#2 " y ! 4#2
!
"1
9
16
−3
−4
−5
(b) Center: "1, #4#
−6
a " 4, b " 3, c " !16 # 9 " !7
−8
Vertices: "1, 0#, "1, #8#
Foci: "1, #4 ± !7 #
e"
!7
c
"
a
4
30. (a) 4"x 2 ! 4x ! 4# ! 25" y 2 # 6y ! 9# " #141 ! 16 ! 225
(c)
y
6
4"x ! 2#2 ! 25" y # 3#2 " 100
"x ! 2#2 " y # 3#2
!
"1
25
4
© Houghton Mifflin Company. All rights reserved.
(b) a " 5, b " 2, c " !21
Center: "#2, 3#
Foci: "#2 ± !21, 3#
!21
5
31. (a) 3"x 2 ! 4x ! 4# ! 8" y 2 # 14y ! 49# " #403 ! 12 ! 392
3"x ! 2#2 ! 8" y # 7#2 " 1
"x ! 2#2 " y # 7#2
!
"1
1&3
1&8
—CONTINUED—
4
2
−8
−6
−4
−2
x
−2
−4
−6
Vertices: "3, 3#, "#7, 3#
e"
867
2
4
1 2 3 4
x
Chapter 9
868
Topics In Analytic Geometry
31. —CONTINUED—
(b) Center: "#2, 7#
!3
a"
3
,b"
(c)
!2
8
4
6
c2 " a2 # b2 "
$#2
Vertices:
$#2
Foci:
±
Eccentricity:
32. (a)
y
!30
1 1
5
# "
⇒ c"
3 8 24
12
±
!3
3
!30
12
%
,7
4
2
−4
!30&12
!10
c
"
"
a
4
!3&3
(b) a "
%
$x # 52%
2
! 20" y ! 3#2 "
5
4
2
Foci:
e"
1
2
3
4
2
1
,b" ,c"
4
x
!54 # 161 "
!19
4
$52, #3%
Vertices:
y
1
!5
Center:
'x # "5&2#(
" y ! 3#
!
"1
"5&4#
"1&16#
2
−1
x
−1
%
25
25
x 2 # 5x !
! 20" y 2 ! 6y ! 9# " #185 !
! 180
4
4
(c)
−2
,7
x 2 ! 20y 2 # 5x ! 120y ! 185 " 0
$
−3
$52
$52
±
±
!5
2
!19
4
%
, #3
%
, #3
!19&4
!95
c
"
"
a
10
!5&2
−1
−2
−3
33. Vertices: "± 5, 0#
34. Vertices: "0, ± 6#
Foci: "± 4, 0#
Passes through "2, 2#
a " 5, c " 4 ⇒ b " 3
Vertical major axis
x2
Center: "0, 0#, a " 6
25
!
y2
9
"1
x2
y2
!
"1
2
b
36
22
22
!
"1
2
b
36
4
1 8
"1# "
b2
9 9
b2 "
36 9
"
8
2
x2
y2
!
"1
9&2 36
© Houghton Mifflin Company. All rights reserved.
−4
Review Exercises for Chapter 9
35. Vertices: "#3, 0#, "7, 0#
36. Vertices: "2, 0#, "2, 4#
Foci: "0, 0#, "4, 0#
Foci: "2, 1#, "2, 3#
Horizontal major axis
Vertical major axis
Center: "2, 0#
Center: "2, 2#
a " 5, c " 2,
a " 2, c " 1,
b " !25 # 4 " !21
b " !4 # 1 " !3
"x # h#2 " y # k#2
!
"1
a2
b2
"x # h# 2 " y # k#2
!
"1
b2
a2
"x # 2#2
y2
!
"1
25
21
"x # 2# 2 " y # 2# 2
!
"1
3
4
38.
37. a " 5, b " 4, c " !a2 # b2 " !25 # 16 " 3
The foci should be placed 3 feet on either side of
the center and have the same height as the pillars.
x2
y2
!
" 1, a " !324 " 18, b " !196 " 14
324 196
Longest distance: 2a " 2"18# " 36 feet
Shortest distance: 2b " 2"14# " 28 feet
c2 " a2 # b 2 " 128
Foci: "± 8!2, 0#
Distance between foci: 16!2 ) 22.63 feet
40. a "
39. a # c " 1.3495 $ 109
a ! c " 1.5045 $ 109
Adding, 2a " 2.854
Then
c " 1.5045 $
10 9
$
109 ⇒ a " 1.427
# 1.427
$
109
$
" 0.0775
$
e"
10 9.
© Houghton Mifflin Company. All rights reserved.
(c)
y
5
4
3
y 2 x2
# "1
4
5
(b) a " 2, b " !5,
c " !4 ! 5 " 3
Center: "0, 0#
Vertices: "0, ± 2#
Foci: "0, ± 3#
Eccentricity "
c
3
"
a 2
c
" 0.2056 ⇒ c " ae " 7.4016
a
x2
y2
!
"1
1296 1241.2
c
e " ) 0.0543.
a
41. (a) 5y 2 # 4x 2 " 20
72
" 36
2
b2 " a2 # c2 " 362 # 7.40162 ) 1241.2
109
1
−5 −4 −3 −2 −1
−3
−4
−5
869
1 2 3 4 5
x
Chapter 9
870
42. (a)
Topics In Analytic Geometry
9
4
3
3
(b) a " , b " ,
2
2
x2
y2
#
"1
"9&4# "9&4#
!
9
"! "
2
x2 # y2 "
(c)
y
5
4
3
2
1
9 9
!
4 4
c"
3
3!2
"
2
!2
−5 − 4 − 3
−1
Foci:
$
±
x
$± 32, 0%
%
3!2
,0
2
Eccentricity "
c
3!2&2
"
" !2
a
3&2
43. (a) 9"x2 # 2x ! 1# # 16" y2 ! 2y ! 1# " 151 ! 9 # 16
(c)
y
6
9"x # 1#2 # 16" y ! 1#2 " 144
4
"x # 1#
" y ! 1#
#
"1
16
9
2
3 4 5
−2
−3
−4
−5
Center: "0, 0#
Vertices:
1
2
2
−6 −4
(b) Center: "1, #1#, a " 4, b " 3, c " 5
2
−2
4
6
8
x
−4
−6
Vertices: "5, #1#, "#3, #1#
−8
Foci: "6, #1#, "#4, #1#
5
4
44. (a) 25" y 2 ! 6y ! 9# # 4"x 2 ! 2x ! 1# " #121 ! 225 # 4
(c)
y
6
4
2
25" y ! 3#2 # 4"x ! 1#2 " 100
" y ! 3#2 "x ! 1#2
#
"1
4
25
(b) Center: "#1, #3#, a " 2, b " 5, c " !29
Vertices: "#1, #1#, "#1, #5#
Foci: "#1, #3 ± !29#
Eccentricity:
!29
2
−8 −6
−2
−4
−6
−8
−10
−12
−14
4 6 8 10
x
© Houghton Mifflin Company. All rights reserved.
Eccentricity:
Review Exercises for Chapter 9
45. (a) " y 2 # 2y ! 1# # 4"x 2 ! 12x ! 36# " #59 ! 1 # 144
" y # 1#2 # 4"x ! 6#2 " #202
"x ! 6#2 " y # 1#2
#
"1
"101&2#
202
(b) Center: "#6, 1#
a2 "
y
30
101 2
101
505
, b " 202, c2 "
! 202 "
2
2
2
Vertices:
Foci:
(c)
$#6 !
±
$
#6 ±
!
Eccentricity: e "
20
10
%
101
,1
2
−30 −20
10
20
30
x
−10
−20
%
505
,1
2
−30
!505
c
"
" !5
a !101
46. (a) 9"x 2 # 8x ! 16# # " y 2 # 8y ! 16# " #119 ! 144 # 16 " 9
9"x # 4#2 # " y # 4#2 " 9
"x # 4#2 #
" y # 4#2
"1
9
(b) Center: "4, 4#
(c)
y
12
a " 1, b " 3, c " !10
10
Vertices: "4 ± 1, 4#: "3, 4#, "5, 4#
8
6
Foci: "4 ± !10, 4#
4
2
Eccentricity: e " !10
−2
© Houghton Mifflin Company. All rights reserved.
47.
x2
y2
#
"1
a2 b2
4
8
10 12
x
48. Vertices: "0, ± 1#
Foci: "0, ± 3#
a"4
c2 " a2 ! b2 ⇒ 36 " 16 ! b2
⇒ b " !20 " 2!5
x2
y2
#
"1
16 20
−2
Vertical transverse axis
Center: "0, 0#
a " 1, c " 3, b " !9 # 1 " !8
y2 x2
# "1
1
8
871
872
Chapter 9
Topics In Analytic Geometry
49. Foci: "0, 0#, "8, 0# ⇒ c " 4
50. Vertical transverse axis
Center: "4, 0#
Center: "3, 0# ⇒ c " 2
Asymptotes:
a
" 2 ⇒ a " 2b
b
b
" 2 ⇒ b " 2a
a
y " ± 2"x # 4# ⇒
a 2 ! b2 " c 2
"2b# 2 ! b 2 " 4
c2 " a2 ! b2
16 " a2 ! "2a#2 " 5a2 ⇒ a "
4
!5
,b"
8
!5
"x # 4#2
y2
#
"1
16&5
64&5
b2 "
4
5
a2 "
16
5
" y # k# 2 "x # h#2
#
"1
a2
b2
5y 2 5"x # 3#2
#
"1
16
4
51. d2 # d1 " 186,000"0.0005#
y
(−100, 0)
100
80
60
40
20
− 60
− 20
2a " 93
a " 46.5
c " 100
B
x
A
− 40
− 60
− 80
−100
b " !c2 # a2
x2
y2
# 2"1
2
a
b
x " 60 ⇒ y2 " b2
(100, 0)
20
$ax
2
2
%
# 1 " "1002 # 46.52#
60
$46.5
(60, 0)
2
2
%
# 1 ) 5211.57 ⇒ y ) 72.2
52. Let the friends be at B and C, you at the origin A. The sound at C is heard 2 seconds after B:
5
5
2279
2
2
2
2a " CD # BD " 2"1100
5280 # " 12 . Thus, a " 24 , c " 2 and b " c # a " 576 . Thus, using miles,
the hyperbola is
x2
y2
#
" 1.
"25&576# "2279&576#
Now place the center at "1, 0# and determine the second hyperbola.
2a " DB # AD " 6
5
⇒ a"
$1100
5280 %
8
c " 1 and b2 " 1 #
25 39
"
64 64
"x # 1#2
y2
#
"1
"25&64#
"39&64#
1.
2.
y
y
1
1
D
C
−2
D
A
−1
B
1
−1
2
x
C
−2
A
−1
B
1
−1
2
x
© Houghton Mifflin Company. All rights reserved.
72.2 miles north
Review Exercises for Chapter 9
53.
873
3x2 ! 2y2 # 12x ! 12y ! 29 " 0
3"x2 # 4x ! 4# ! 2" y2 ! 6y ! 9# " #29 ! 12 ! 18
3"x # 2#2 ! 2" y ! 3#2 " 1
Ellipse
54. 4x 2 ! 4y 2 # 4x ! 8y # 11 " 0
A " C " 4 ⇒ Circle
55.
56. #4y 2 ! 5x ! 3y ! 7 " 0
5x 2 # 2y 2 ! 10x # 4y ! 17 " 0
5"x 2 ! 2x ! 1# # 2" y 2 ! 2y ! 1# " #17 ! 5 # 2
A " 0, C " #4, AC " 0
5"x ! 1#2 # 2" y ! 1#2 " #14
Parabola
" y ! 1#2 "x ! 1#2
#
"1
7
"14&5#
Hyperbola
57. xy # 4 " 0
y
A " 0, B " 1, C " 0
cot 2& "
x"
!2
2
5
4
3
2
y′
A#C
'
"0 ⇒ &"
B
4
!2
"x% # y% #, y "
2
−3 −2
x′
x
2 3 4 5
−2
"x% ! y% #
xy " 4
!2
2
"x% # y% #
!2
2
"x% ! y% # " 4
1
1
"x% #2 # " y% #2 " 4
2
2
© Houghton Mifflin Company. All rights reserved.
"x% #2 " y% #2
#
"1
8
8
Hyperbola
58. x 2 # 10xy ! y 2 ! 1 " 0
y
A " C " 1 ⇒ cot 2& " 0 ⇒ & "
x"
!2
2
"x% # y% #, y "
* 22 "x% # y% #+
2
!
!2
2
y′
'
4
1
x
"x% ! y% #
1
* 22 "x% # y% #+* 22 "x% ! y% #+ ! * 22 "x% ! y% #+
# 10
!
!
!
2
!1"0
1
1
1
1
"x% #2 ! " y% #2 # x%y% # 5""x% #2 # " y% #2# ! "x% #2 ! x%y% ! " y% #2 " #1
2
2
2
2
#4"x% #2 ! 6" y% #2 " #1
"x% #2 " y% #2
#
"1
1&4
1&6
Hyperbola
x′
2
2
Chapter 9
874
Topics In Analytic Geometry
59. 5x2 # 2xy ! 5y2 # 12 " 0
y
y′
A " 5, B " #2, C " 5
cot 2& " 0 ⇒ & "
x"
!2
2
2
'
4
1
−3
!2
"x% # y% #, y "
x′
3
2
−2
−1
"x% ! y% #
1
−1
x
2
3
−2
−3
5x2 # 2xy ! 5y2 " 12
* 22"x% # y% #+
!
5
2
* 12"x% #
2
5
* 22"x% # y% #+* 22"x% ! y% #+ ! 5* 22"x% ! y% #+
!
#2
!
+
!
2
*
" 12
+
1
1
1
# x%y% ! " y% #2 # "x% #2 ! " y% #2 ! 5 "x% #2 ! x%y% ! " y% #2 " 12
2
2
2
4"x% #2 ! 6" y% #2 " 12
"x% #2 " y% #2
!
"1
3
2
Ellipse
60. cot 2& "
x"
!2
2
y
4#4
'
"0 ⇒ &"
8
4
!2
"x% # y% #, y "
2
y′
4
2
"x% ! y% #
−4 −3
*
4
!2
2
+ *
2
"x% # y% #
!8
!2
2
"x% # y% #
x′
3
!2
2
+ *
"x% ! y% # !4
! 7!2
!2
2
*
3
4
x
2
"x% ! y% #
!2
2
+
2
−4
+
*
"x% # y% # ! 9!2
!2
2
+
"x% ! y% # " 0
2'"x% #2 ! " y% #2 # 2x%y% ( ! 4'"x% #2 # " y% #2( ! 2'"x% #2 ! " y% #2 ! 2x%y% ( ! 7"x% # y% # ! 9"x% ! y% # " 0
8"x% #2 ! 16x% ! 2y% " 0
Parabola
61. (a) B2 # 4AC " "#8#2 # 4"16#"1# " 0
62. (a) B2 # 4AC " #300 ⇒ Ellipse
(b) 7y 2 # 8xy ! "13x 2 # 45# " 0
Parabola
(b) y2 ! "5 # 8x#y ! "16x2 # 10x# " 0
y"
(c)
"8x # 5# ± !"5 # 8x#2 # 4"16x2 # 10x#
2
y"
(c)
8x ± !"64x 2# # 4"7#"13x 2 # 45#
14
4
2
−5
2
−6
6
−4
−10
© Houghton Mifflin Company. All rights reserved.
y% " #4"x% #2 # 8x%
Review Exercises for Chapter 9
63. (a) B2 # 4AC " "2#2 # 4"1#"1# " 0
(c)
875
10
Parabola
(b) y2 ! "2x # 2!2 #y ! "x2 ! 2!2x ! 2# " 0
−15
"2!2 # 2x# ± !"2x # 2!2#2 # 4"x2 ! 2!2x ! 2#
y"
0
0
2
64. (a) B2 # 4AC " 100 # 4
65. Adding the equations,
" 96 > 0 ⇒ Hyperbola
(b)
y2
# 10xy ! "
x2
24x ! 240 " 0 ⇒ x " #10. Then:
! 1# " 0
4"100# ! y2 # 560 # 24y ! 304 " 0
y"
10x ± !100x2 # 4"x2 ! 1#
2
y"
10x ± !96x2 # 4
2
y2 # 24y ! 144 " 0
" y # 12#2 " 0 ⇒ y " 12
Solution: "#10, 12#
y " 5x ± !24x2 # 1
(c)
4
−6
6
−4
66. 4x 2 ! 4y 2 " 100
67.
9x # 4y 2 " 0
Adding:
4x 2
t
#2
#1
0
1
2
3
x
#8
#5
#2
1
4
7
y
15
11
7
3
#1
#5
! 9x " 100
y
4x 2 ! 9x # 100 " 0
16
© Houghton Mifflin Company. All rights reserved.
"x # 4#"4x ! 25# " 0 ⇒ x " 4, # 254
12
If x " 4:
8
4y 2 " 9"4# " 36
4
−12
y2 " 9 ⇒ y " ± 3
−8
−4
8
−4
25
2
If x " # 25
4 , 9"# 4 # " 4y , impossible.
Answer: "4, 3#, "4, #3#
68. x " !t, y " 8 # t
y
8
t
0
1
2
3
4
7
x
0
1
!2
!3
2
5
y
8
7
6
5
4
3
6
4
2
1
−3 −2 −1
−1
x
1
2
4
5
6
x
Chapter 9
876
Topics In Analytic Geometry
69. x " 5t # 1, y " 2t ! 5
70. x " 4t ! 1, y " 8 # 3t
t " 15 "x ! 1# ⇒
t " 14"x # 1# ⇒
y " 25 "x ! 1# ! 5 " 25 x ! 27
5 , line
y " 8 # 3"14"x # 1## " # 34x ! 35
4
y
y
20
40
15
30
20
5
10
−10
−10
10
20
x
−20 −15 −10 −5
x
30
5 10
−10
−15
−20
−20
71. x " t2 ! 2, y " 4t2 # 3
72. x " ln 4t, y " t 2
1 2x
ex " 4t, t " 14 e x ⇒ y " "14 e x# " 16
e
2
t2 " x # 2 ⇒
y " 4"x # 2# # 3 " 4x # 11,
x ≥ 2
y
4
y
3
12
2
10
1
8
−4 −3 −2 −1
6
4
−2
2
−3
−4 −2
20
6
8
2
3
4
−4
x
4
x
1
10 12
−4
1
73. x " t3, y " t2
2
4
74. x " , y " t 2 # 1
t
y
5
1
t " x1&3 ⇒ y " x2&3
2
y
20
16
4
t" ⇒ y" 2 #1
x
x
4
3
2
1
1
2
3
4
x
−2
−12
−8
−3
3 t
75. x " !
77. x "
76. x " t
3 t " !
3 x " x1&3
y"!
y"t
t " x3 ⇒ y " t " x3
−4
y"t
2
−3
x
4
1
t
y"t"
y " x3
−4
3
4
1
x
4
−2
−6
6
−4
−6
6
−4
8
12
© Houghton Mifflin Company. All rights reserved.
− 4 − 3 −2 − 1
−1
Review Exercises for Chapter 9
y"
80. x " t 2, y " !t
79. x " 2t
78. x " t
1 1
"
t
x
4
−6
y " 4t
t " y2
y " 2"2t# " 2x
x " " y 2# 2 " y 4, y ≥ 0
Line
4
y "!
x, x ≥ 0
3
4
6
−6
−4
877
6
−1
5
−1
−4
81. x " 1 ! 4t
y " 2 # 3t
t"
$
%
x#1
x#1
3
3
⇒ y"2#3
"2# x!
4
4
4
4
y"
11 3
# x
4
4
6
−4
8
−2
3x ! 4y # 11 " 0
82. x " t ! 4, y " t 2
83. x " 3
7
t"x#4
y"t
y " "x # 4# 2
−2
4
−3
−1
−4
84. x " t
85. x " 6 cos &, y " 6 sin &
7
x
y
cos & " , sin & "
6
6
y"2
−6
6
x2
y2
!
"1
36 36
© Houghton Mifflin Company. All rights reserved.
−1
x2 ! y2 " 36
86. x " 3 ! 3 cos &, y " 2 ! 5 sin &
cos & "
87. y " 6x ! 2
x#3
y#2
, sin & "
3
5
"x # 3#
" y # 2#
!
"1
9
25
2
2
9
Vertical line: x " 3
10
x " t, y " 6t ! 2
x " #t, y " #6t ! 2
8
−4
Other answers possible
14
−4
8
−12
12
−8
878
Chapter 9
Topics In Analytic Geometry
89. y " x 2 ! 2
88. x " t, y " 10 # t
x " #t, y " 10 ! t
x " t, y " t2 ! 2
Many answers possible
x " t ! 1, y " "t ! 1#2 ! 2 " t2 ! 2t ! 3
Other answers possible
91. x " x1 ! t"x2 # x1# " 3 ! t"8 # 3# " 5t ! 3
90. x " t, y " 2t3 ! 5t
x " #t, y " #2t # 5t
y " y1 ! t" y2 # y1# " 5 ! t "0 # 0# " 5
Many answers possible
or x " t, y " 5
3
92. x " x1 ! t "x2 # x1# " 2 ! t "2 # 2# " 2
93. x " x1 ! t "x2 # x1#
y " y1 ! t " y2 # y1#
" #1 ! t '10 # "#1#( " 11t # 1
" #1 ! t "4 # "#1## " #1 ! 5t
y " y1 ! t " y 2 # y1# " 6 ! t "0 # 6# " #6t ! 6
or x " 2, y " t
5
5
94. x " x1 ! t "x2 # x1# " 0 ! t " 2 # 0# " 2 t
95. "90, 4# is on the curve:
y " y1 ! t" y2 # y1# " 0 ! t "6 # 0# " 6t
90 " 0.82v0 t ⇒ v0 "
or x " 5t, y " 12t
90
0.82t
90
* 0.82t
+t # 16t
4 " 7 ! 0.57
96. From Exercise 95, v0 " 54.23.
⇒
16t 2 " 3 !
0.57"90#
⇒ t ) 2.024
0.82
Hence, v0 )
90
) 54.23 ft&sec.
0.82"2.024#
97. From Exercise 96:
x " 0.82"54.23#t " 44.47t
2
98. From Exercise 95,
t ) 2.024 seconds.
25
" 7 ! 30.91t # 16t 2
0
100
0
The maximum height is approximately 21.9 feet for t ) 0.97.
99.
$1, '4 %
100.
π
2
(−5, − π3 (
π
2
(1, 4π (
1 2 3 4 5
1
2
3
0
0
$1, # 74'%, $#1, 54'%, $#1, # 34'%
$#5, 53'%, $5, 23'%, $5, # 43'%
© Houghton Mifflin Company. All rights reserved.
y " 7 ! 0.57"54.23#t # 16t 2
Review Exercises for Chapter 9
$
101. "r, &# " #2, #
11'
6
%
$
102. "r, &# " 1,
π
2
)−2 , − 116π )
1
2
3
5π
6
0
1
$#2, '6 %, $2, 76'%, $2, # 56'%
103.
%
π
2
( )
1,
5'
6
879
2
0
3
$#1, # '6 %, $#1, 116'%, $1, # 76'%
$!5, # 43'%
104.
(
10, 3π
4
)
π
2
π
2
( 5, − 43 (
π
1
1
2
2
3
%$
%$
2'
'
5'
!5,
, # !5, # , # !5,
3
3
3
$
7'
6
$
5!3 5
,
2 2
105. "r, &# " 5, #
"x, y# " #
$
%
%
106.
© Houghton Mifflin Company. All rights reserved.
%$
%$
5'
'
7'
, #!10, # , #!10,
4
4
4
π
2
%
π
2
1 2 3 4 5
0
(−4, 23π (
(5, − 76π (
1 2 3
0
$
"r, &# " #4,
$
2'
3
"x, y# " #4 cos
107.
0
0
3
"r, &# " !10, #
$
4
$2, # 53'%
π
2
x " r cos & " 2
)2, − 53π )
$%
1
"1
2
$ 2 %"
y " r sin & " 2
"x, y# " "1, !3 #
!3
1
!3
2
3
4
0
%
%
2'
2'
" "2, #2!3 #
, #4 sin
3
3
%
880
108.
Chapter 9
Topics In Analytic Geometry
$#1, 116'%
$
109. "r, &# " 3,
$ 23 %
!
x " r cos & " #1
!3 1
,
2
2
(
3, 3π
4
%
−1,
11 π
6
%
π
2
(
1
)
1
$ '2 %,
3'
3'
#3!2 3!2
"
, 3 sin
,
4
4
2
2
π
2
)
110. "r, &# " 0,
% $
$
$ 12%
$
%
"x, y# " 3 cos
y " r sin & " #1 #
"x, y# " #
3'
4
2
3
2
0
0
3
111. "x, y# " "0, #9#
the origin
"x, y# " "0, 0#
$
3'
'
, #9,
2
2
3
6
"r, &# " 9,
π
2
%$
%
y
3
(0, 2π (
−3
1 2 3 4 5
9
12
x
−3
0
−6
(0, −9)
−9
−12
112. "x, y# " "#3, 4#
#4
3
"5, 126.87(#, "#5, 306.87(#
or
"5, 2.214#, "#5, 5.356#
(radians)
(−3, 4)
$
"r, &# " 5!2,
3
2
−4 −3 −2 −1
−1
1
2
3
4
y
x
2
−2
1
−3
−2 −1
−1
−4
1
2
3
4
−2
−4
(5, −5)
−5
−6
114. "x, y# " "#3, # !3 #
y
4
3
7'
Third quadrant, & "
6
2
1
" "#3# ! 3 " 12 ⇒ r " 2!3
2
$
"r, &# " 2!3,
%$
%
1
−3
r2
%$
7'
3'
, #5!2,
4
4
7'
'
, #2!3,
6
6
%
−4 −3 −2 −1
−1
(−3, −
3(
−2
−3
−4
1
2
3
4
x
5
6
x
© Houghton Mifflin Company. All rights reserved.
r " 5, tan & "
113. "x, y# " "5, #5#
y
4
Review Exercises for Chapter 9
115. x 2 ! y 2 " 9
116. x2 ! y2 " 20
117. x 2 ! y 2 # 4x " 0
r2 " 9
r2 " 20
r 2 # 4r cos & " 0
r " 2!5
r"3
119.
120.
r " 6 sin &
"r cos &#"r sin &# " #2
" 5 csc &
121.
r2 " 6r sin &
xy " #2
"r cos &#"r sin &# " 5
r2
118. x2 ! y2 " 6y
r " 4 cos &
xy " 5
r2 " #2 sec & csc &
) sec &
122.
4x 2 ! y " 1
2x 2 ! 3y2 " 1
4"r cos &#2 ! "r sin &#2 " 1
2"r cos &#2 ! 3"r sin &#2 " 1
4r 2 cos2 & ! r 2"1 # cos2 &# " 1
r 2"2 cos2 & ! 3 sin2 &# " 1
r 2 '3 cos2 & ! 1( " 1
r 2"2"1 # sin2 &# ! 3 sin2 &# " 1
r2 "
3
r 2"2 ! sin2 &# " 1
1
&!1
cos2
r2 "
123.
126.
881
r"5
124.
125.
r " 12
2
r " 3 cos &
2
x 2 ! y 2 " 5 2 " 25
x ! y " 144
Circle
Circle
1
2 ! sin2 &
r 2 " 3r cos &
x 2 ! y 2 " 3x
127.
r " 8 sin &
r 2 " cos 2&
r2 " 8r sin &
r 2 " 1 # 2 sin2 &
x2 ! y2 " 8y
r 4 " r 2 # 2r 2 sin2 &
"x 2 ! y 2#2 " x 2 ! y 2 # 2y 2
© Houghton Mifflin Company. All rights reserved.
"x 2 ! y 2#2 # x 2 ! y 2 " 0
128.
129. & "
r2 " sin &
5'
6
r3 " r sin &
tan & "
"x2 ! y2#3&2 " y or
"x2 ! y2#3 " y2
y"#
131. r " 5, circle
130. & "
y
1
"#
x
!3
!3
3
tan & "
133. & "
132. r " 3, circle
π
2
π
2
2
4
6
0
y
" !3
x
y " !3x
line
x,
4'
3
'
, y-axis
2
π
2
1
2
4
0
1
2
3
0
882
Chapter 9
134. & " #
Topics In Analytic Geometry
5'
, line
6
136. r " 2 sin &, circle
135. r " 5 cos &, circle
π
2
π
2
π
2
1
2
3
1
0
4
2
3
137. r " 5 ! 4 cos &
4
0
6
1
2
3
0
π
2
Dimpled limaçon
Symmetric with respect to polar axis
r is maximum at & " 0: "r, &# " "9, 0#
2
4
6
8 10 12
0
r * 0 (No zeros)
138. r " 1 ! 4 sin &
π
2
Limaçon with inner loop
Symmetric with respect to & "
'
2
,r, is a maximum at & " 2 : $5, 2 %
'
'
1
r " 0 when 4 sin & " #1 ⇒ sin & " #
4
0
π
2
1 2 3
5
0
'
2
,,
Maximum r -value: r " 8 when & "
Zeros: r " 0 when & ) 0.6435, 2.4981
3'
2
$sin & " 35%
140. r " 2 # 6 cos &
π
2
Limaçon with inner loop
Symmetry: polar axis
,,
Maximum: r " 8 when & " '
1
Zero: r " 0 when cos & " 3 ⇒ & ) 1.231, 5.052
2
0
© Houghton Mifflin Company. All rights reserved.
Limaçon with loop
,,
3
1
⇒ & ) 3.394, 6.031
4
139. r " 3 # 5 sin &
Symmetry: line & "
2
Review Exercises for Chapter 9
141. r " #3 cos 2&, 0 ≤ & ≤ 2'
883
π
2
Four-leaved rose
'
, polar axis, and pole
2
'
3'
The value of ,r, is a maximum (3) at & " 0, , ', .
2
2
' 3' 5' 7'
r " 0 for & " , , ,
4 4 4 4
Symmetric with respect to & "
0
4
142. r " cos 5&
π
2
Five-leaved rose
Symmetric with respect to polar axis
,r, is maximum value of 1 at & "
r " 0 for & "
n'
, n " 0, 1, 2, . . .
5
0
1
'
2n'
!
, n " 0, 1, 2, . . .
10
10
143. r 2 " 5 sin 2&
π
2
Lemniscate
Symmetry with respect to pole
,,
Maximum r -value: !5 when & "
' 5'
,
4 4
1
2
0
3
'
3'
Zeros: r " 0 when & " 0, , ',
2
2
144. r2 " cos 2&
π
2
Lemniscate
© Houghton Mifflin Company. All rights reserved.
Symmetry: Pole, polar axis, and line & "
'
2
1
,,
0
Maximum: r " 1 when & " 0, ', 2'
Zeros: r " 0 when & "
145. r "
2
1 # sin &
' 3' 5' 7'
, , ,
4 4 4 4
146. r "
e"1
1
,e"2
1 ! 2 sin &
Hyperbola symmetric with & " '&2 and having
vertices at "1&3, '&2# and "#1, 3'&2#
Parabola
3
6
−6
6
−2
−3
3
−1
884
Chapter 9
147. r "
"
e"
Topics In Analytic Geometry
4
5 # 3 cos &
4&5
1 # "3&5# cos &
148. r "
2
−3
6
#6
"
#1 ! 4 cos & 1 # 4 cos &
Hyperbola "e " 4#
3
4
−2
3
5
−6
6
Ellipse
149. r "
"
e"
−4
5
6 ! 2 sin &
150. r "
2
5&6
1 ! "1&3# sin &
−3
3
Parabola "e " 1#
4
−2
1
3
−4
Ellipse
152. Parabola: r "
4
1 # cos &
ep
,e"1
1 ! e sin &
$ 2%
Vertex: 2,
Vertical directrix: x " #4
'
Focus: "0, 0# ⇒ p " 4
r"
153. Ellipse: r "
8
−4
151. e " 1
r"
3
3&4
"
4 # 4 cos & 1 # cos &
4
1 ! sin &
ep
1 # e cos &
Vertices: "5, 0#, "1, '# ⇒ a " 3
e"
c
2
2&3 p
5
" ,5"
⇒p"
a 3
1 # "2&3# cos 0
2
r"
"2&3#"5&2#
5&3
5
"
"
1 # "2&3# cos & 1 # "2&3# cos & 3 # 2 cos &
154. Hyperbola: r "
ep
1 ! e cos &
Vertices: "1, 0#, "7, 0# ⇒ a " 3
One focus: "0, 0# ⇒ c " 4
e"
c
4
4&3 p
7
" ,1"
⇒p"
1
!
"
4&3
#
cos
0
4
a 3
r"
7
"4&3#"7&4#
7&3
"
"
1 ! "4&3# cos & 1 ! "4&3# cos & 3 ! 4 cos &
© Houghton Mifflin Company. All rights reserved.
One focus: "0, 0# ⇒ c " 2
Review Exercises for Chapter 9
885
155. e " 0.093
Use r "
2a "
r"
ep
.
1 # e cos &
0.093p
0.093p
!
" 0.1876p " 3.05 ⇒ p ) 16.258, ep ) 1.512
1 # 0.093 cos 0 1 # 0.093 cos '
1.512
1 # 0.093 cos &
1.512
) 1.383 astronomical units
1 ! 0.093
Perihelion:
Aphelion:
156. Use r "
e"1
1.512
) 1.667 astronomical units
1 # 0.093
ep
1 # e sin &
(parabola)
When & "
#'
, r " 6,000,000.
2
p
r"
1 # sin
r"
(horizontal directrix below pole).
$#2'%
"
p
" 6,000,000 ⇒ p " 12,000,000
2
12,000,000
1 # sin &
'
When & " # , distance is approximately 6,430,781 miles.
3
157. False. The y 4-term is not
second degree.
158. False. There are many sets
possible. For example,
x " t, y " 3 # 2t
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x " 3t, y " 3 # 6t.
160. (a) Major axis horizontal
(b) Circle
159. (a) Vertical translation
(b) Horizontal translation
(c) Reflection in the y-axis
(d) Parabola opens more slowly.
161. The number b must be less than 5. The ellipse
becomes more circular and approaches a circle
of radius 5.
(c) Ellipse is flatter.
(d) Horizontal translation
162. The orientation of the graph would be reversed.
163. (a) The speed would double.
(b) The elliptical orbit would be flatter. The length
of the major axis is greater.
886
Chapter 9
Chapter 9
Topics In Analytic Geometry
Practice Test
1. Find the vertex, focus and directrix of the parabola x2 ! 6x ! 4y " 1 # 0.
2. Find an equation of the parabola with its vertex at !2, !5" and focus at !2, !6".
3. Find the center, foci, vertices, and eccentricity of the ellipse x2 " 4y2 ! 2x " 32y " 61 # 0.
4. Find an equation of the ellipse with vertices !0, ± 6" and eccentricity e # 12.
5. Find the center, vertices, foci, and asymptotes of the hyperbola 16y2 ! x2 ! 6x ! 128y " 231 # 0.
6. Find an equation of the hyperbola with vertices at !± 3, 2" and foci at !± 5, 2".
7. Rotate the axes to eliminate the xy-term. Sketch the graph of the resulting equation, showing both sets of axes.
5x2 " 2xy " 5y2 ! 10 # 0
8. Use the discriminant to determine whether the graph of the equation is a parabola, ellipse, or hyperbola.
(a) 6x2 ! 2xy " y2 # 0
(b) x2 " 4xy " 4y2 ! x ! y " 17 # 0
For Exercises 9 and 10, eliminate the parameter and write the corresponding rectangular equation.
9. x # 3 ! 2 sin $, y # 1 " 5 cos $
10. x # e2t, y # e4t
11. Convert the polar point !#2, !3%"$4" to rectangular coordinates.
12. Convert the rectangular point !#3, !1" to polar coordinates.
14. Convert the polar equation r # 5 cos $ to rectangular form.
15. Sketch the graph of r # 1 ! cos $.
16. Sketch the graph of r # 5 sin 2$.
17. Sketch the graph of r #
3
.
6 ! cos $
18. Find a polar equation of the parabola with its vertex at !6, %$2" and focus at !0, 0".
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13. Convert the rectangular equation 4x ! 3y # 12 to polar form.