Hyperbola: Conjugate Axis

Hyperbola: Conjugate Axis

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

Formula: Conjugate Axis of x2/a2 - y2/b2 = 1

For the hyperbola x^2/a^2 - y^2/b^2 = 1, the conjugate axis is 2b.

The conjugate axis is the green segment.

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

the conjugate axis is the height of the green box
formed by
the vertices and the asymptotes of a hyperbola.

Hyperbola - Asymptotes

The x2 term is (+).

So the conjugate axis is 2b.

Transverse axis - Formula (x2/a2 - y2/b2 = 1)

Example 1: Conjugate Axis 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.

The x2 term is (+).

So the conjugate axis is 2⋅4.

2⋅4 = 8

So (conjugate axis) = 8.

This is the graph of x2/32 - y2/42 = 1.

The x2 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 y2/b2 - x2/a2 = 1

For the hyperbola x^2/a^2 - y^2/b^2 = 1, the conjugate axis is 2b.

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

the conjugate axis is the width of the green box
formed by
the vertices and the asymptotes of a hyperbola.

The y2 term is (+).

So the conjugate axis is 2a.

Transverse axis - Formula (y2/b2 - x2/a2 = 1)

Example 2: Conjugate Axis 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.

The y2 term is (+).

So the conjugate axis is 2⋅2.

2⋅2 = 4

So (conjugate axis) = 4.

This is the graph of y2/22 - x2/22 = 1.

The y2 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.