Avon High School
Section: 10.2
ACE COLLEGE ALGEBRA II - NOTES
The Hyperbola
Mr. Record: Room ALC-129
Semester 2 - Day 23
Definition of a Hyperbola
If you’ve ever turned on a lamp and looked at the light that eminates from the top and
bottom of the lamp shade, then you have probably seen our next conic section we’ll
discuss, the hyperbola.
Notice how the graph containes two disjoint branches. Although each part may look like
a parabola, neither branch features the characteristics of a parabola.
A hyperbola is the set of all points, P, in a plane in
which the difference of the distances from two fixed
points F1 and F2 is constant. These two fixed points are,
again, called the foci. The midpoint of the segment
connecting the foci is the center of the hyperbola.
Standard Form of the Hyperbola
Standard Forms of the Equation of the Hyperbola
The standard form of the equation of a hyperbola with center ( h, k ) is
( x  h) 2 ( y  k ) 2

1
a2
b2
Transverse axis is horizontal
Asymptotes:
yk 
b
b
 x  h  and y  k    x  h 
a
a
or
( y  k ) 2 ( x  h) 2

1
a2
b2
Transverse axis is vertical
Asymptotes: y  k  a  x  h  and y  k   a  x  h 
b
b
The vertices are a units from the center and the foci are c units from the center, with c 2  a 2  b 2 .
Both hyperbolas shown to the right are
centered at the origin.
Example 1
Graphing an Hyperbola Centered at the Origin
Graph and find the foci, vertices and
x2 y 2
equations of the asymptotes for

 1.
9
4
y




x


















Example 2
Graphing a Hyperbola Centered at the Origin
Graph and find the foci, vertices and
equations of the asymptotes for 6 y 2  24 x 2  6 .
y




x









Example 3
Finding the Equation of a Hyperbola from Its Foci and Vertices
Find the standard from of the equation of a hyperbola with foci at (0, 5) and (0,5) and
vertices (0, 3) and (0,3) .

Example 4
Graphing a Hyperbola Centered at (h, k)
( x  3) 2 ( y  1) 2

 1 and find
4
1
the foci, vertices and equations of the
asymptotes.
Graph
Example 5
Writing an Equation of a Hyperbola in Standard Form
Convert the equation of the hyperbola 4 x 2  9 y 2  24 x  90 y  153  0 into standard form
and determine the location of the foci and the equationsof the asymptotes.
Applications
The hyperbolic shape of a sonic
boom.
The trajectories of asteroids and comets.
Example 6
An Application Involving Hyperbolas
An explosion is recorded by two microphones that are 2 miles apart. Microphone M 1 received the sound 4
seconds before microphone M 2 . Assuming sound travels at 1100 feet per second, determine the possible
locations of the explosion relative to the location of the microphones.