Lesson8.3

Hyperbolas

Precalculus

2 2

1. Find the distance between the points ( , ) and ( ,4).

2. Solve for in terms of . 1 16 2

Solve for algebraically.

3. 3 12 3 8 10

a b c

y xy x

x

x x

Quick Review

2 2( ) (4 )c a b

2 216 8y x 2 28 16y x 216 8y x

3 12 10 3 8x x

2

3 12 10 3 8x x 3 12 100 20 3 8 3 8x x x

20 100 20 3 8x

1 5 3 8x 4 3 8x No Solution

2 2

2 2

Solve for algebraically. 4. 6 12 6 1 1

5. Solve the system of equations:

2

16

x x x

c a

ac a

c

Quick Review

2c a

2 2 1( )2

6

2

aaa

a

28 12 4 16a a a

164 4

2

aa

a

24 4 8 0a a

2 26 12 1 6 1x x

22 26 12 1 6 1x x 2 2 26 12 1 2 6 1 6 1x x x

212 2 6 1x 236 6 1x

237 6x37

6x 222

6

No Solution since it can never equal zero

What you’ll learn about

Geometry of a HyperbolaTranslations of HyperbolasEccentricity and OrbitsReflective Property of a HyperbolaLong-Range Navigation

… and whyThe hyperbola is the least known conic section, yet it is used astronomy, optics, and navigation.

Hyperbola

A hyperbola is the set of all points in a plane whose distances from two fixed points in the plane have a constant difference.

The fixed points are the foci of the hyperbola. The line through the foci is the focal axis. The point on the focal axis midway between the foci is the center. The points where the hyperbola intersects its focal axis are the vertices of the hyperbola.

Hyperbola with Center (0,0)

2 2

2 2 Standard equation 1

Focal axis -axis

Foci ( ,0)

Vertices ( ,0)

Semitransverse axis

x y

a bx

c

a

2 2 2

Semiconjugate axis

Pythagorean relation

Asymptotes

a

b

c a b

by x

a

Hyperbola with Center (0,0)

2 2

2 2 Standard equation 1

Focal axis -axis

Foci (0, )

Vertices (0, )

Semitransverse axis

y x

a by

c

a

2 2 2

Semiconjugate axis

Pythagorean relation

Asymptotes

a

b

c a b

by x

a

Example Finding the Vertices and Foci of an Hyperbola

2 2Find the vertices and the foci of the hyperbola 9 4 36.x y

Thus the vertices are ( 2,0), and the foci are ( 13,0).

2 2

Divide both sides by 36 to put the equation in standard form.

14 9

x y

2 2 2 2 2So 4, 9, and 13.a b c a b

Example Finding an Equation of an Hyperbola

2 2 2Thus 16 1 15. a c b

The center is (0,0). The foci are on the -axis with 4.

The semiconjugate axis is 2 / 2 1.

y c

b

2 2

Thus the equation of the hyperbola is 1.15 1

y x

Find an equation of the hyperbola with foci (0,4) and (0, 4)

whose conjugate axis has length 2.

Hyperbola with Center (h,k)

2 2

2 2

Standard equation

1

Focal axis

Foci ( , )

Vertices ( , )

Sem

x h y k

a by k

h c k

h a k

2 2 2

imajor axis

Semiminor axis

Pythagorean relation

Asymptotes ( )

a

b

c a b

by x h k

a

Hyperbola with Center (h,k)

2 2

2 2

Standard equation

1

Focal axis

Foci ( , )

Vertices ( , )

Semima

y k x h

a bx h

h k c

h k a

2 2 2

jor axis

Semiminor axis

Pythagorean relation

Asymptotes ( )

a

b

c a b

ay x h k

b

Example Locating Key Points of a Hyperbola

The center is at ( 1,0).

22

Find the center, vertices, and foci of the hyperbola

1 1.

4 9

x y

Because the semitransverse axis 4 2,

the vertices are at ( , ) 1 2,0 or ( 3,0) and (1,0).

a

h a k

2 2Because 4 9 13, the foci are at

( , ) ( 1 13,0) or approximately (2.61,0) and ( 4.61,0).

c a b

h c k

Eccentricity of an Hyperbola

2 2

The of a hyperbola is ,

where is the semitransverse axis, is the

semiconjugate axis, and is the distance from the

center to either focus.

c a bea a

a b

c

eccentricity

Homework:

Text pg663/664 Exercises

# 2-42 (intervals of 4)And #52