Hyperbola: Asymptote

How to find the asymptote of a hyperbola: formula, 2 examples, and their solutions.

Formula: x2/a2 - y2/b2 = 1

Formula

The asymptote is a line
the graph follows.

So, for a hyperbola,
there are two asymptotes.

Hyperbola: Equation

For the horizontal hyperbola
x2/a2 - y2/b2 = 1,

the asymptotes are
y = ±[b/a]x.

Example 1

Example

Solution

9 = 32
16 = 42

x2/32 - y2/42 = 1

a = 3
b = 4

Then the asymptotes are
y = ±[4/3]x.

So y = ±[4/3]x is the answer.

Formula: y2/a2 - x2/b2 = 1

Formula

For the vertical hyperbola
y2/a2 - x2/b2 = 1,

the asymptotes are
y = ±[a/b]x.

a and b are switched.

Example 2

Example

Solution

To make the right side 1,
divide both sides by 9.

9 = 32

-x2
= -x2/1
= -x2/12

y2/32 - x2/12 = 1

a = 3
b = 1

Then the asymptotes are
y = ±[3/1]x.

x

3/1 = 3

So y = ±3x is the answer.