Hyperbolas9.3

Definition of a Hyperbola

A hyperbola is the set of all points (x, y) in a plane, the difference of whose distances from two distinct fixed points (foci) is a positive constant.

Standard Form Equation of a Hyperbola

The standard form of the equation of a hyperbola with center at (h, k) is

Horizontal

Vertical

The vertices are a units from the center, and the foci are c units from the center. c2 = a2 + b2.

1

2

2

2

2

b

ky

a

hx

1

2

2

2

2

b

hx

a

ky

Example 1

Find the standard form of the equation of the hyperbola with foci (-1, 2) and (5, 2) and vertices (0, 2) and (4, 2).

Example 2

Sketch the hyperbola whose equation is 4x2 – y2 = 16.

Example 3

Find the standard form of the equation of 9x2 – y2 – 36x – 6y + 18 = 0.