Identify the center, vertices, and asymptotes of a hyperbola from its equation Use the eccentricity and focal length to write the equation of a hyperbola.

1. 10.3 Hyperbolas
Chapter 10 Analytic Geometry

2. Concepts and Objectives
⚫ Hyperbolas
⚫ Identify equations of hyperbolas
⚫ From the equation, identify the center, direction of
opening, vertices, x-radius, y-radius, slope of the
asymptotes, and foci
⚫ Sketch the hyperbola
⚫ Determine the eccentricity
⚫ Write the equation of the hyperbola

3. Hyperbolas
⚫ Hyperbolas have two disconnected branches. Each
branch approaches diagonal asymptotes.
⚫ Parts of a hyperbola:
⚫ Center
⚫ Vertices
⚫ Asymptotes
⚫ Hyperbola • ••

4. Hyperbolas
⚫ The general equation of a hyperbola is
or
⚫ The hyperbola opens in whichever direction has the
positive term (x-direction if x is positive, y-direction if y
is positive).
⚫ The slope of the asymptotes is always .
⚫ The vertices are rx or ry from the center, whichever is
positive. a is the positive term radius, b is the negative
term radius.
  − −
− =    
   
22
1
x y
x h y k
r r
  − −
− + =    
   
22
1
x y
x h y k
r r
 y
x
r
r

5. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y

6. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y
( ) ( ) ( ) ( )++ − =+ +−− −22 2 22 2
9 10 8 1975 9 54 4 4 4x x y y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

7. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y
( ) ( ) ( ) ( )++ − =+ +−− −22 2 22 2
9 10 8 1975 9 54 4 4 4x x y y
( ) ( )+ − − = −
2 2
9 64 45 3x yNotice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

8. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y
( ) ( ) ( ) ( )++ − =+ +−− −22 2 22 2
9 10 8 1975 9 54 4 4 4x x y y
( ) ( )+ − −
− =
− − −
2 2
9 5 4 4 36
36 36 36
x yNotice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

9. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y
( ) ( ) ( ) ( )++ − =+ +−− −22 2 22 2
9 10 8 1975 9 54 4 4 4x x y y
( ) ( )+ − −
− =
− − −
2 2
9 5 4 4 36
36 36 36
x y
( ) ( )+ −
− + =
2 2
5 4
1
4 9
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

10. Hyperbolas
⚫ Example: Graph − + + + =2 2
9 4 90 32 197 0x y x y
( ) ( ) ( ) ( )++ − =+ +−− −22 2 22 2
9 10 8 1975 9 54 4 4 4x x y y
( ) ( )+ − −
− =
− − −
2 2
9 5 4 4 36
36 36 36
x y
( ) ( )+ −
− + =
2 2
5 4
1
4 9
x y
( ) ( )+ −
− + =
2 2
2 2
5 4
1
2 3
x y
Notice that the
negative sign has
been factored as
well!
Remember to
distribute the
negative sign!

11. Hyperbolas
⚫ Example: Graph
Center (–5, 4)
− + + + =2 2
9 4 90 32 197 0x y x y
( ) ( )+ −
− + =
2 2
2 2
5 4
1
2 3
x y

12. Hyperbolas
⚫ Example: Graph
Center (–5, 4)
opens in y-direction
rx = 2, ry = 3
 vertices 3
− + + + =2 2
9 4 90 32 197 0x y x y
( ) ( )+ −
− + =
2 2
2 2
5 4
1
2 3
x y

13. Hyperbolas
⚫ Example: Graph
Center (–5, 4)
opens in y-direction
rx = 2, ry = 3
 vertices 3
slope of asymptotes:
− + + + =2 2
9 4 90 32 197 0x y x y
( ) ( )+ −
− + =
2 2
2 2
5 4
1
2 3
x y

3
2

14. Hyperbolas
⚫ Example: Graph
Center (–5, 4)
opens in y-direction
rx = 2, ry = 3
 vertices 3
slope of asymptotes:
− + + + =2 2
9 4 90 32 197 0x y x y
( ) ( )+ −
− + =
2 2
2 2
5 4
1
2 3
x y

3
2

15. Focal Length
⚫ In an ellipse, the sum of the distances from a point on the
ellipse to the two foci is constant, but in a hyperbola, it’s
the difference between the distances that is constant.
⚫ To find the focal radius, we can use the Pythagorean
Theorem.
⚫ Notice that c > a for the
hyperbola. a
b
c
•
= +2 2 2
c a b

16. Eccentricity
⚫ Like the ellipse, the eccentricity of the hyperbola
determines the basic shape, and like the ellipse, the
eccentricity of the hyperbola is
⚫ In an ellipse, e will always be between 0 and 1, but in a
hyperbola, e will always be greater than 1.
=
c
e
a

17. Eccentricity
⚫ Example: Find the eccentricity of the hyperbola
− =
2 2
1
9 4
x y

18. Eccentricity
⚫ Example: Find the eccentricity of the hyperbola
a = 3, b = 2
− =
2 2
1
9 4
x y
− =
2 2
2 2
1
3 2
x y
= +2 2 2
3 2c
= + =9 4 13
= 13c
= 
13
1.2
3
e

19. Eccentricity
⚫ Example: Write the equation of the hyperbola with
eccentricity 2 and foci at (–9, 5) and (–3, 5).

20. Eccentricity
⚫ Example: Write the equation of the hyperbola with
eccentricity 2 and foci at (–9, 5) and (–3, 5).
The foci’s coordinates tell us that the hyperbola opens in
the x-direction, that the center is at (–6, 5), and that c = 3.
=
3
2
a
 =1.5a
= − =2
9 2.25 6.75b
=2
2.25a
( ) ( )+ −
− =
2 2
6 5
1
2.25 6.75
x y

21. Classwork
⚫ College Algebra
⚫ Page 978: 10-22 (even), page 968: 16-28 (even),
page 959: 52-56 (even)