# Hyperbola: Transverse Axis

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

## Definition

A hyperbola is the set of points

whose difference of the distances from the foci

is constant.

Hyperbola - Proof of the formula *x*^{2}/*a*^{2} - *y*^{2}/*b*^{2} = 1

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

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

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

Then the vertices are (±*a*, 0).

So the transverse axis (= distance between the vertices)

is 2*a*.

## Example 1: Transverse 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 transverse axis is 2⋅3.

2⋅3 = 6

So (transverse axis) = 6.

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

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

So the transverse axis is 6.

And the it is parallel to the *x*-axis.

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

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

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

Then the vertices are (0, ±*b*, 0).

So the transverse axis (= distance between the vertices)

is 2*b*.

## Example 2: Transverse 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 transverse axis is 2⋅2.

2⋅2 = 4

So (transverse axis) = 4.

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

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

So the transverse axis is 4.

And it is parallel to the *y*-axis.