Hyperbola: Transverse Axis

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.

A hyperbola is the set of points
whose difference of the distances from the foci
is constant.

Hyperbola - Proof of the formula x2/a2 - y2/b2 = 1

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

For the hyperbola x^2/a^2 - y^2/b^2 = 1, the transverse axis (= distance between the vertices) is 2a.

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

the x2 term is (+).

Then the vertices are (±a, 0).

So the transverse axis (= distance between the vertices)
is 2a.

Example 1: Transverse Axis of x2/9 - y2/16 = 1

Find the transverse of the given parabola. 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 transverse axis is 2⋅3.

2⋅3 = 6

So (transverse axis) = 6.

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

The x2 term is (+).

So the transverse axis is 6.

And the it is parallel to the x-axis.

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

For the hyperbola y^2/b^2 - x^2/a^2 = 1, the transverse axis (= distance between the vertices) is 2b.

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

the y2 term is (+).

Then the vertices are (0, ±b, 0).

So the transverse axis (= distance between the vertices)
is 2b.

Example 2: Transverse Axis of x2 - y2 = -4

Find the transverse of the given parabola. 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 transverse axis is 2⋅2.

2⋅2 = 4

So (transverse axis) = 4.

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

The y2 term is (+).

So the transverse axis is 4.

And it is parallel to the y-axis.