Ellipse: Major Axis

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.

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 diameter) is 2a.

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

Find the major axis of the given ellipse. x^2/25 + y^2/16 = 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 x^2/a^2 + y^2/b^2 = 1, if a < b, then the major axis (= longest diameter) is 2b.

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

Find the major axis of the given ellipse. 9x^2 + 4y^2 = 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.