# 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 *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1,

if *a* > *b*,

then the major axis (= longest diamter)

is the horizontal diameter.

So the major axis is 2*a*.

Minor axis - Formula (*a* > *b*)

## Example 1: Major Axis of *x*^{2}/25 + *y*^{2}/16 = 1

Change the equation in standard form.

25 = 5^{2}

16 = 4^{2}

So *x*^{2}/5^{2} + *y*^{2}/4^{2} = 1.

*a* > *b* (5 > 4)

So the major axis is 2⋅5.

2⋅5 = 10

So (major axis) = 10.

This is the graph of *x*^{2}/5^{2} + *y*^{2}/4^{2} = 1.*a* > *b*

So the major axis, 10,

is the horizontal diameter.

## Formula: If *a* < *b*

For the ellipse *x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1,

if *a* < *b*,

then the major axis (= longest diamter)

is the vertical diameter.

So the major axis is 2*b*.

Minor axis - Formula (*a* < *b*)

## Example 2: Major Axis of 9*x*^{2} + 4*y*^{2} = 36

Change the equation in standard form.

Divide both sides by 36.

9/36 = 1/4

4/36 = 1/9

4 = 2^{2}

9 = 3^{2}

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

*a* < *b* (2 < 3)

So the major axis is 2⋅3.

2⋅3 = 6

So (major axis) = 6.

This is the graph of *x*^{2}/2^{2} + *y*^{2}/3^{2} = 1.*a* < *b*

So the major axis, 6,

is the vertical diameter.