# Hyperbola: Conjugate Axis

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

## Formula: Conjugate Axis of *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1

The conjugate axis is the green segment.

For the hyperbola *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1,

the conjugate axis is the height of the green box

formed by

the vertices and the asymptotes of a hyperbola.

Hyperbola - Asymptotes

The *x*^{2} term is (+).

So the conjugate axis is 2*b*.

Transverse axis - Formula (*x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1)

## Example 1: Conjugate Axis of *x*^{2}/9 - *y*^{2}/16 = 1

Change the equation in standard form.

9 = 3^{2}

16 = 4^{2}

So *x*^{2}/3^{2} - *y*^{2}/4^{2} = 1.

The *x*^{2} term is (+).

So the conjugate axis is 2⋅4.

2⋅4 = 8

So (conjugate axis) = 8.

This is the graph of *x*^{2}/3^{2} - *y*^{2}/4^{2} = 1.

The *x*^{2} term is (+).

So the conjugate axis is 8.

It's the height of the green box

formed by

the vertices and the asymptotes of a hyperbola.

## Formula: Conjugate Axis of *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1

For the hyperbola *y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1,

the conjugate axis is the width of the green box

formed by

the vertices and the asymptotes of a hyperbola.

The *y*^{2} term is (+).

So the conjugate axis is 2*a*.

Transverse axis - Formula (*y*^{2}/*b*^{2} - *x*^{2}/*a*^{2} = 1)

## Example 2: Conjugate Axis of *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 = 2^{2}

So *y*^{2}/2^{2} - *x*^{2}/2^{2} = 1.

The *y*^{2} term is (+).

So the conjugate axis is 2⋅2.

2⋅2 = 4

So (conjugate axis) = 4.

This is the graph of *y*^{2}/2^{2} - *x*^{2}/2^{2} = 1.

The *y*^{2} term is (+).

So the conjugate axis is 4.

It's the width of the green box

formed by

the vertices and the asymptotes of a hyperbola.