# 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 *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 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) = 2*b*.

Hyperbola: formula, foci, transverse axis

If the hyperbola is *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 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.*x*^{2}/3^{2} - *y*^{2}/4^{2} = 1.

Then *a* = 3 and *b* = 4.

*y*^{2} 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 *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 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) = 2*a*.

Hyperbola: formula, foci, transverse axis

If the hyperbola is *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 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.*y*^{2}/2^{2} - *x*^{2}/2^{2} = 1.

Then *a* = *b* = 2.

*x*^{2} 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.