Hyperbola: Asymptotes

Hyperbola: Asymptotes

How to find the asymptotes of a hyperbola: formulas, examples, and their solutions.

Formula: Asymptotes of x2/a2 - y2/b2 = 1

For the hyperbola x^2/a^2 - y^2/b^2 = 1, the equations of the asymptotes are y = +-(b/a)x.

The asymptote is a line the graph follows.

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

the equations of the asymptotes are
y = ±(b/a)x.

Example 1: Asymptotes of x2/9 - y2/16 = 1

Find the conjugate axis of the given hyperbola. x^2/9 - y^2/16 = 1

Change the equation in standard form.

9 = 32
16 = 42

So x2/32 - y2/42 = 1.

a = 3
b = 4

So the asymptotes are
y = ±(4/3)x.

This is the graph of the hyperbola x2/32 - y2/42 = 1
and its asymptotes y = ±(4/3)x.

You can see that
as x goes to ∞ or -∞,
the hyperbola follows the asymptotes.

Formula: Asymptotes of y2/b2 - x2/a2 = 1

For the hyperbola y^2/b^2 - x^2/a^2 = 1, the equations of the asymptotes are also y = +-(b/a)x.

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

the equations of the asymptotes are also
y = ±(b/a)x.

Example 2: Asymptotes of x2 - y2 = -4

Find the conjugate axis of the given hyperbola. x^2 - y^2 = -4

Change the equation in standard form.

Divide both sides by -4.

Switch the order of the terms
in the left side.

4 = 22

So y2/22 - x2/22 = 1.

b = 2
a = 2

So the asymptotes are
y = ±(2/2)x.

2/2 = 1

So the asymptotes are y = ±x.

This is the graph of the hyperbola y2/22 - x2/22 = 1
and its asymptotes y = ±x.

You can see that
as x goes to ∞ or -∞,
the hyperbola follows the asymptotes.