* Locate a hyperbola’s vertices and foci. * Write equations of hyperbolas in standard form. * Graph hyperbolas centered at the origin. * Graph hyperbolas not centered at the origin. * Solve applied problems involving hyperbolas.

1. 8.2 The Hyperbola
Chapter 8 Analytic Geometry

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Locate a hyperbola’s vertices and foci.
⚫ Write equations of hyperbolas in standard form.
⚫ Graph hyperbolas centered at the origin.
⚫ Graph hyperbolas not centered at the origin.
⚫ Solve applied problems involving hyperbolas.

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
term is positive. a is the positive term radius, b is the
negative term radius.
 
 
− −
− =
 
   
   
2
2
1
x y
x h y k
r r
 
 
− −
− + =
 
   
   
2
2
1
x y
x h y k
r r
 y
x
r
r

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

6. Hyperbolas
⚫ Example: Graph − + + + =
2 2
9 4 90 32 197 0
x y x y
( ) ( ) ( ) ( )
+
+ − =
+ +
−
− −
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x 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 0
x y x y
( ) ( ) ( ) ( )
+
+ − =
+ +
−
− −
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
( ) ( )
+ − − = −
2 2
9 6
4 4
5 3
x y
Notice 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 0
x y x y
( ) ( ) ( ) ( )
+
+ − =
+ +
−
− −
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x x y y
( ) ( )
+ − −
− =
− − −
2 2
9 5 4 4 36
36 36 36
x y
Notice 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 0
x y x y
( ) ( ) ( ) ( )
+
+ − =
+ +
−
− −
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x 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 0
x y x y
( ) ( ) ( ) ( )
+
+ − =
+ +
−
− −
2
2 2 2
2 2
9 10 8 197
5 9 5
4 4 4 4
x 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 0
x 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 0
x 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 0
x 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 0
x 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. Classwork
⚫ College Algebra 2e
⚫ 8.2: 12-44 (×4); 8.1: 32-56 (×4); 7.8: 36-44 (×4)
⚫ 8.2 Classwork Check
⚫ Quiz 8.1