Section 9-5

Hyperbolas

Objectives

• I can write equations for hyperbolas

• I can graph hyperbolas

• I can Complete the Square to obtain Standard Format of the equation

Hyperbola Definition

• A hyperbola is the set of all points in a plane such that the absolute value of the difference of the distances from any point on the hyperbola to the foci is a constant.

Basic Diagram

Transverse Axis F1 F2

Conjugate Axis

Vertex

Equation Information

• Transverse Axis = 2a units

• Conjugate Axis = 2b units

• Vertex is a units from the center point (h, k)

• Focus Point is c units from center point

Equations• For Hyperbolas with Foci

at (-c, 0) and (c, 0) and center point (h, k) opening left and right (horizontal transverse axis)

• For Hyperbolas with Foci at (0, -c) and (0, c) opening up and down (vertical transverse axis)

1)()(

2

2

2

2

b

ky

a

hx1

)()(2

2

2

2

b

hx

a

ky

• Where c2 = a2 + b2 • Where c2 = a2 + b2

Hyperbolas

Example 1

2 2( 2) ( 1)1

25 16

x y

Example 2

2 2( 4) ( 1)1

9 49

x y

Example 3

2 2( 1)1

4 1

y x

Example 4

2 2( 5) ( 3)1

16 9

y x

Example 5

2 2( 5) ( 3)1

25 36

y x

Example 1• Write the equation for the hyperbola with

transverse axis length 8 units and foci at (6,0) and (-6,0)

• Based on foci the transverse axis is horizontal

• Since length = 2a = 8; then a = 4

• Then c2 = a2 + b2

• 62 = 42 + b2

• 36 = 16 + b2

• b2 = 20

1)()(

2

2

2

2

b

ky

a

hx

120

)0(

16

)0( 22

yx

Draw the graph: 125

)5(

16

)2( 22

yx

a2 = 16, so a = 4

b2 = 25, so b = 5

c2 = a2 + b2

c2 = 16 + 25 = 41

c = 6.4

Center Point (-2, 5)

Transverse axis is horizontal = 2a = 8 units

Asymptotes y = +/- 5/4 x

EXAMPLE 1 Graph an equation of a hyperbola

Graph 25y2 – 4x2 = 100. Identify the vertices, foci, and asymptotes of the hyperbola.

SOLUTION

STEP 1

Rewrite the equation in standard form.

25y2 – 4x2 = 100 Write original equation.

25y2

100 – 4x2

100100 100= Divide each side by 100.

y2

4 –y2

25 = 1 Simplify.

Complete the square

• 4y2 - 2x2 + 16y – 4x - 10 = 0

• (4y2 + 16y) +(-2x2 - 4x) = 10

• 4(y2 + 4y) -2(x2 + 2x) = 10

• 4(y2 + 4y + 4) -2(x2 + 2x + 1) = 10 + 16 – 2

• 4(y + 2)2 – 2(x + 1)2 = 24

• Now divide all terms by 24

2 2( 2) ( 1)1

6 12

y x

Homework

• Worksheet 10-8