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Hyperbola

See how to solve a hyperbola
(transverse axis, foci, equation, asymptotes).
6 examples and their solutions.

Definition

A hyperbola is the set of points that satisfy
|PF - PF'| = (constant, transverse axis).
F, F': foci

Hyperbola: x2a2 - y2b2 = 1

Equation

x2a2 - y2b2 = 1



a2 + b2 = c2

Transverse Axis: 2a
Foci: (c, 0), (-c, 0)

Example

x29 - y216 = 1
1. Transverse axis?
2. Foci?
Solution

Example

Foci: (3, 0), (-3, 0)
Transverse axis: 4
Equation of the hyperbola?
Solution

Asymptotes

x2a2 - y2b2 = 1



Asymptotes: y = ±bax
The graph of a hyperbola follows two asymptotes.
(purple dashed lines)
Asymptotes: y = [b/a]x, y = -[b/a]x

Example

x29 - y216 = 1
Asymptotes?
Solution

Hyperbola: x2a2 - y2b2 = -1

Equation

x2a2 - y2b2 = 1



a2 + b2 = c2

Transverse Axis: 2b
Foci: (0, c), (0, -c)

Example

4x2 - y2 = -4
1. Transverse axis?
2. Foci?
Solution

Example

Foci: (0, 5), (0, -5)
Transverse axis: 6
Equation of the hyperbola?
Solution

Asymptotes

x2a2 - y2b2 = -1



Asymptotes: y = ±bax
The asymptotes formula is the same
as the upper hyperbola.
Asymptotes: y = [b/a]x, y = -[b/a]x

Example

y2 - 9x2 = 9
Asymptotes?
Solution