Ellipse: Foci

Ellipse: Foci

How to find the foci of an ellipse: formulas, examples, and their solutions.

Formula: If a > b

For the ellipse x^2/a^2 + y^2/b^2 = 1, if a > b, then the foci are (+-c, 0) and c^2 = a^2 - b^2.

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

if a > b,
then the foci are (±c, 0)
and c2 = a2 - b2.

Example 1: Foci of x2/25 + y2/16 = 1

Find the foci 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 = 5
b = 4
And a > b.

So c2 = 52 - 42.

52 = 25
42 = 16

25 - 16 = 9

So c2 = 9.

Square root both sides.

Then c = ±3.

c = ±3
And a > b.

So the foci are (±3, 0).

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

c = ±3
And a > b. (5 > 4)

So the foci are (±3, 0).

Formula: If a < b

For the ellipse x^2/a^2 + y^2/b^2 = 1, if a < b, then the foci are (0, +-c) and c^2 = b^2 - a^2.

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

if a < b,
then the foci are (0, ±c)
and c2 = b2 - a2.

Example 2: Foci of 9x2 + 4y2 = 36

Find the foci 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 = 2
b = 3
And a < b.

So c2 = 32 - 22.

32 = 9
22 = 4

9 - 4 = 5

So c2 = 5.

Square root both sides.

Then c = ±√5.

c = ±√5
And a < b.

So the foci are (0, ±√5).

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

c = ±√5
And a < b. (2 < 3)

So the foci are (0, ±√5).