# Hyperbola: Asymptotes, Conjugate Axis

How to find the equations of the asymptotes and the conjugate axis of the hyperbola: definition, formula, proof, examples, and their solutions.

## Definition

The asymptotes of a hyperbola are the lines
that the graph of the hyperbola follows,
as x and y become large.

## Formula 1

If x2/a2 - y2/b2 = 1,
then the conjugate axis is the height of the dashed box
which is formed by the asymptotes
and the vertieces of the hyperbola.

(conjugate axis) = 2b.

Hyperbola: formula, foci, transverse axis

If the hyperbola is x2/a2 - y2/b2 = 1,
then the equation of the asymptotes are
y = ±(b/a)x.

## Example 1

To find a, b, and c,
change the equation into the standard form.

x2/32 - y2/42 = 1.

Then a = 3 and b = 4.

y2 term is (-).

So (transverse axis) = 2⋅3 = 6.

Hyperbola: formula, foci, transverse axis

Then (conjugate axis) = 2⋅4 = 8.

So the asymptotes are y = ±(4/3)x.

You can see that
the graph of the hyperbola
follows its asymptotes y = ±(4/3)x.

And the transverse axis and the conjugate axis
perpendicularly bisect each other.

## Formula 2

If y2/b2 - x2/a2 = 1,
then the conjugate axis is the width of the dashed box
which is formed by the asymptotes
and the vertieces of the hyperbola.

(conjugate axis) = 2a.

Hyperbola: formula, foci, transverse axis

If the hyperbola is y2/b2 - x2/a2 = 1,
then the equation of the asymptotes are
y = ±(b/a)x.
[not y = ±(a/b)x]

## Example 2

To find a, b, and c,
change the equation into the standard form.

y2/22 - x2/22 = 1.

Then a = b = 2.

x2 term is (-).

So (transverse axis) = 2⋅2 = 4.
(2 from b)

Hyperbola: formula, foci, transverse axis

Then (conjugate axis) = 2⋅2 = 4.
(2 from a)

So the asymptotes are y = ±(2/2)x.

You can see that
the graph of the hyperbola
follows its asymptotes y = ±(2/2)x.

And the transverse axis and the conjugate axis
perpendicularly bisect each other.