OHHS

Pre-Calculus

Mr. J. Focht

8.3 Hyperbolas

• Geometry of a Hyperbola

• Translations of Hyperbolas

• Eccentricity

8.3

A Hyperbola is a Conic Section

8.3

Hyperbola Definition

• Set of all points whose difference of the distances to 2 fixed points is constant.

Focus Focus

(x,y)

d1

d2

d1 – d2 is the same for whatever (x,y) you choose on the blue curves.

8.3

Hyperbola Terms

Focus FocusCenter

Transverse Axis

Conjugate Axis

Vertex

Vertex

Asymptote

Asymptote

Focal Axis

8.3

Hyperbola Terms

Focus FocusCenter

Asymptote

Asymptote

a = distance from center to vertex

a

b

b = distance from vertex to asymptote

c = distance from center to focus

c

c2 = a2 + b2

8.3

Hyperbola Equation

Focus FocusCenter

Asymptote

Asymptote

a

bc

(h,k)

1)()(

2

2

2

2

b

ky

a

hx

8.3

Hyperbola Equation

Fo

cus

Fo

cus

Asy

mpt

ote

Asy

mpt

ote

ab

c (h,k

)

1)()(

2

2

2

2

b

hx

a

ky

8.3

Asymptote Equations

Focus FocusCenter

Asymptote

Asymptote

a

bc

(h,k)

)( hxa

bky

8.3

Asymptote Equations

Fo

cus

Fo

cus

Asy

mpt

ote

Asy

mpt

ote

ab

c (h,k

)

)( hxb

aky

8.3

Example

• Find the vertices and the foci of the hyperbola 4x2 - 9y2 = 36.

• Divide by sides by 36.

1)()(

2

2

2

2

b

ky

a

hx

2 24 9 36

36 36 36

x y

2 2

19 4

x y

8.3

Example: Find the Center, Vertices, and Foci

2 2

19 4

x y

a2 = 9

a = 3

Vertices (-3, 0),

(3, 0)

(h, k) = (0, 0)

b2 = 4

c2 = a2 + b2 = 9 + 4 = 13

Foci 0,13

13,0

8.3

Now You Try

1716

22

yx

Find the center, vertices, and foci

8.3

Find the equation of the hyperbola

4

(1,-5)

(1, 1)

The center is halfway between the foci.

(1, -2)2

2

a

1)()(

2

2

2

2

b

hx

a

ky 1+2

4

c = distance from center to a focus

c = 3

c2 = a2 + b2

9 = 4 + b2

b2 = 5

5

8.3

Now You Try

• P. 663, #23: Find the equation that satisfies these conditions: Foci (±3, 0), transverse axis length 4

8.3

Example

• Find the coordinates of the center, foci, and vertices, and the equations of the asymptotes of the graph of

4x2 – y2 + 24x + 4y + 28 = 0

4x2 + 24x – y2 + 4y = -28

4(x2 + 6x ) - (y2 -4y ) = -28

4(x+3)2 – (y-2)2 = 4

+ 9+ 36

+ 4

- 4

8.3

Example

4(x+3)2 – (y-2)2 = 4

14

2)(y

1

3)(x 22

a = 1 b = 2

c2 = a2 + b2

c2 = 1 + 4

c2 = 5

5c Now let’s find the vertices, foci, and asymptotes on the graph.

8.3

Example

5c 1

4

2)(y

1

3)(x 22

a=1 b=2(-3, 2)

(-2, 2)(-4, 2)

2),53(

2),53(

h)(xa

bky 3)(x

1

22y

8.3

Now You Try

• P. 664, #49: Find the center, vertices, and foci of 9x2 – 4y2 – 36x + 8y – 4 =0

8.3

Eccentricity

• Hyperbolas have eccentricities too.

a

cE Since c > a, E > 1

8.3

Example

• Write the equation of the hyperbola with center at (-2, -4) , a focus of (2,-4) and eccentricity

3

4

1b

4)(y

a

2)(x2

2

2

2

c = distance from center to focus = 4

a

c

3

4E Since c= 4, a = 3

c2 = a2 + b2 42 = 32 + b2 b2 = 7

9 7

8.3

Now You Try

• p. 663, #37: Find an equation in standard form for the hyperbola that satisfies the conditions: Center(-3,6), a=5, e=2, vertical transverse axis

8.3

Home Work

• P. 663-665, #2, 6, 24, 28, 32, 38, 44, 50, #63-68

8.3