9.5 Notes – Hyperbolas

Hyperbolas: the set of all points for which the difference of the distances to two foci is a constant.

d1

d2 – d1= constant

(x, y)

focus focus

d2

center

The imaginary line between the focal points is the ‘transverse’ axis of the hyperbola.

transverse

asymptote

focus focus

(c, 0)

ca

vertex

Horizontal transverse axis

asymptotefocus

(c, 0)

vertex ca

Vertical transverse axis

A hyperbola can be graphed by locating the vertices (using the a distance from the center) and drawing the two asymptotes through the center of the hyperbola. The foci can be located by using the formula: .

c 2 a2 b2

Standard Form of equation for a hyperbola(note the a2 is always in the lead term)

Horizontal Transverse axis Vertical Transverse axis

Asymptote: Asymptote:

Foci:

ba b

a

(x h)2

a2 (y k)2

b2 1

(y k)2

a2 (x h)2

b2 1

c 2 a2 b2

by x h k

a a

y x h kb

Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.

x 2

9y 2

161

a)

horizontal

Center: (0, 0)4

3y x

by x h k

a

40 0

3y x

Ex 1: State if the hyperbola is horizontal/vertical, find the center, and the eqn of asymptotes.

b)

vertical

Center: (-2, 1)

(y 1)2

49

(x 2)2

91

ay x h k

b

72 1

3y x

7 17 7 11

3 3 3 3y x y x

7 141

3 3y x

7 141

3 3y x

7 72 1 2 1

3 3y x y x

Ex 2: Graph each hyperbola by filling in the missing information

2 2

14 25

x y a)

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

0 0y = 0

2 0 -2 0

c 2 a2 b22 2

14 25

x y

2 4 25c 2 29c

29 5.4c

Ex 2: Graph each hyperbola by filling in the missing information

2 2

14 25

x y a)

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

0 0y = 0

5.45.4 0 -5.4 0

2 0 -2 0

2 2

14 25

x y b

y x h ka

250 0

4y x

25

4y x

Ex 2: Graph each hyperbola by filling in the missing information

2 2

14 25

x y a)

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

0 0y = 0

5.4-5.4 0 5.4 0

25

4y x

2 0 -2 0

2 22 36 2 36y x

Ex 2: Graph each hyperbola by filling in the missing information

2 22 36 2 36y x

2 22 2

136 1

y x

36 36 36

Ex 2: Graph each hyperbola by filling in the missing information

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

-2 2x = -2

-2 8 -2 -4

2 22 2

136 1

y x

c 2 a2 b2

2 36 1c 2 37c

37 6.1c

2 22 2

136 1

y x

Ex 2: Graph each hyperbola by filling in the missing information

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

-2 2x = -2

-2 8 -2 -4

2 22 2

136 1

y x

6.1-2 8.1 -2 -4.1

ay x h k

b

62 2

1y x

2 22 2

136 1

y x

6 2 2y x 6 2 2y x

6 12 2y x 6 12 2y x

6 14y x 6 10y x

Ex 2: Graph each hyperbola by filling in the missing information

Horizontal or Vertical

center: ( , )

transverse axis(eq):

vertices: ( , ) ( , )

c = ______

foci: ( , ) ( , )

Asymp:

-2 2x = -2

-2 8 -2 -4

2 22 2

136 1

y x

6.1-2 8.1 -2 -4.1

6 14y x 6 10y x

Ex 3: Write the equation of the hyperbola centered at the origin with foci (-4, 0) (4, 0) and vertices (-3, 0) and (3, 0)

(x h)2

a2 (y k)2

b2 1

2 2

2

( 0) ( 0)1

9

x y

b

c 2 a2 b2

2 2 24 3 b 216 9 b

27 b

2 2

19 7

x y

Ex 4: Write the equation of the hyperbola centered at the origin with foci (0, 2) (0, -2) and vertices (0, 1) and (0, -1)

2 2

2 2

( ) ( )1

y k x h

a b

2 2

2

( 0) ( 0)1

1

y x

b

c 2 a2 b2

2 2 22 1 b 24 1 b

23 b

2 2

11 3

y x

Ex 5: Write the eqn of the hyperbola with center (-2, 1), vertices at (-2, 5) and (-2, -3) and a b-value of 8.

2 2

2 2

( ) ( )1

y k x h

a b

2 2( 1) ( 2)1

16 64

y x

(-2, 1)

(-2, 5)

(-2, -3)

Ex 6: Write the equation in standard form: 2 29 8 54 56 0x y x y

2 28 9 54 56x x y y

2 28 ___ 9 6 ___ 56 ___ ___x x y y 16 –169 81

2 24 9 3 9x y

2 24 3

19 1

x y

2 23 4

11 9

y x