# Ellipse: Major Axis

How to find the major axis of an ellipse: definition, formulas, examples, and their solutions.

## Definition

An ellipse is the set of points
whose sum of the distances from the foci
is constant.

Ellipse - Proof of the formula (a > b)

## Formula: If a > b

For the ellipse x2/a2 + y2/b2 = 1,

if a > b,
then the major axis (= longest diamter)
is the horizontal diameter.

So the major axis is 2a.

Minor axis - Formula (a > b)

## Example 1: Major Axis of x2/25 + y2/16 = 1

Change the equation in standard form.

25 = 52
16 = 42

So x2/52 + y2/42 = 1.

a > b (5 > 4)

So the major axis is 2⋅5.

2⋅5 = 10

So (major axis) = 10.

This is the graph of x2/52 + y2/42 = 1.

a > b

So the major axis, 10,
is the horizontal diameter.

## Formula: If a < b

For the ellipse x2/a2 + y2/b2 = 1,

if a < b,
then the major axis (= longest diamter)
is the vertical diameter.

So the major axis is 2b.

Minor axis - Formula (a < b)

## Example 2: Major Axis of 9x2 + 4y2 = 36

Change the equation in standard form.

Divide both sides by 36.

9/36 = 1/4

4/36 = 1/9

4 = 22
9 = 32

So x2/22 + y2/32 = 1.

a < b (2 < 3)

So the major axis is 2⋅3.

2⋅3 = 6

So (major axis) = 6.

This is the graph of x2/22 + y2/32 = 1.

a < b

So the major axis, 6,
is the vertical diameter.