Section 9.2

The Hyperbola

Overview

• In Section 9.1 we discussed the ellipse, one of four conic sections.

• Now we continue onto the hyperbola, which in itself is unusual because to obtain the graph you must intersect two cones instead of one.

The Hyperbola

• A hyperbola is the set of all points in the plane the difference of who distances from two fixed points, called foci, is constant.

• The line through the foci intersects the hyperbola in two points, called vertices.

• The line segment that joins the vertices is called the transverse axis, and the midpoint of the transverse axis is the center of the hyperbola.

Pictures

Case I: Center at the Origin

• There are two possible sub-cases when the hyperbola is centered at the origin:

1. The foci and vertices are on the x-axis (the transverse axis is horizontal).

2. The foci and vertices are on the y-axis (the transverse axis is vertical).

• In either case:• a represents the distance from the center to a

vertex.• c represents the distance from the center to a

focus.

Case I (continued)

• The standard form of the equation of a hyperbola with center at the origin and horizontal transverse axis is:

12

2

2

2

b

y

a

x

Case I (continued)

• The standard form of the equation of a hyperbola with center at the origin and vertical transverse axis is:

12

2

2

2

b

x

a

y

Case I (continued)

• Important relationship:

222 acb

Case I (continued)

• Hyperbolas have asymptotes!!• When the transverse axis is horizontal, the

equations of the asymptotes are

xa

by

xa

by

Asymptotes (continued)

• When the transverse axis is vertical, the equations of the asymptotes are

xb

ay

xb

ay

Pictures

Hyperbolas not centered at the origin

• The standard form of the equation of a hyperbola with horizontal transverse axis is:

1

2

2

2

2

b

ky

a

hx

Continued

• The standard form of the equation of a hyperbola with vertical transverse axis is:

1

2

2

2

2

b

hx

a

ky

Asymptotes

• For hyperbola with a horizontal transverse axis, the equations of the asymptotes are

hxa

bky

hxa

bky

Asymptotes (continued)

• For hyperbola with a vertical transverse axis, the equations of the asymptotes are

hxb

aky

hxb

aky

Pictures

Examples

• Find the vertices, locate the foci, and give the equations of the asymptotes:

1

64

4

49

2

13649

22

22

yx

yx

More Examples—Draw The Picture!

• Write the equation of the hyperbola:

1.Foci at (0, -8) and (0, 8); vertices at (0, 1) and (0, -1)

2.Center (3, -2); focus (8, -2); vertex (7, -2)

One More…

• Convert the equation to standard form by completing the square on x and y.

01112450425 22 yxyx