# Ellipse: Formula

How to write the equation of an ellipse by using its foci and major axis: formulas, examples, and their solutions.

## Formula: If *a* > *b*

If the vertices of the ellipse are (±*a*, 0), (0, ±*b*)

and *a* > *b*,

then the equation of the ellipse is*x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1.

## Example 1: Foci (4, 0) and (-4, 0), Major Axis 10, Ellipse?

The foci are (4, 0) and (-4, 0).

Then *c* = 4.

Ellipse - Foci

The major axis is 10.

And the *y* values of the foci are the same.

So the major axis is the horizontal diameter: 2*a*.

So 2*a* = 10.

Divide both sides by 2.

Then *a* = 5.

*a* = 5*c* = 4

The major axis is 2*a*.

So *a* > *b*.

So 5^{2} - *b*^{2} = 4^{2}.

5^{2} = 25

4^{2} = 16

Move 25 to the right side.

Then -*b*^{2} = -9.

Multiply both sides by -1.

Then *b*^{2} = 9.

Instead of finding the value of *b*,

directly use *b*^{2} = 9

to write the equation of the ellipse.

*a* = 5*b*^{2} = 9

So the equation of the ellipse is*x*^{2}/5^{2} + *y*^{2}/9 = 1.

5^{2} = 25

So [*x*^{2}/25 + *y*^{2}/9 = 1] is the answer.

This is the graph of *x*^{2}/25 + *y*^{2}/9 = 1.

Its foci are (4, 0) and (-4, 0).

And its major axis is 2⋅5 = 10.

## Example 2: Foci (0, 1) and (4, 1), Major Axis 6, Ellipse?

Lightly draw the given conditions.

It says the foci are (0, 1) and (4, 1).

And the major axis is 6.

So draw an ellipse like this.

The foci are (0, 1) and (4, 1).

And the distance between the foci is 2*c*.

So 2*c* = 4 - 0.

4 - 0 = 4

Divide both sides by 2.

Then *c* = 2.

*c* = 2

And the *y* values of the foci are the same.

So the original foci are (-2, 0) and (2, 0).

But the given foci are (0, 1) and (4, 1).

So the foci are under a translation.

Use (2, 0) and (4, 1) to find the translation:

(4, 1) = (2 + 2, 0 + 1).

So the translation is

(*x*, *y*) → (*x* + 2, *y* + 1).

Translation of a point

Next, it says the major axis is 6.

And the *y* values of the foci are the same.

So the major axis is the horizontal diameter: 2*a*.

So 2*a* = 6.

Divide both sides by 2.

Then *a* = 3.

*a* = 3*c* = 2

The major axis is 2*a*.

So *a* > *b*.

So 3^{2} - *b*^{2} = 2^{2}.

Ellipse - Foci (*a* > *b*)

3^{2} = 9

2^{2} = 4

Move 9 to the right side.

Then -*b*^{2} = -5.

Multiply both sides by -1.

Then *b*^{2} = 5.

Instead of finding the value of *b*,

directly use *b*^{2} = 5

to write the equation of the ellipse.

*a* = 3*b*^{2} = 5

The ellipse is under the translation

(*x*, *y*) → (*x* + 2, *y* + 1).

So the equation of the ellipse is

(*x* - 2)^{2}/3^{2} + (*y* - 1)^{2}/5 = 1.

Translation of a function

3^{2} = 9

So (*x* - 2)^{2}/9 + (*y* - 1)^{2}/5 = 1.

This is the graph of (*x* - 2)^{2}/9 + (*y* - 1)^{2}/5 = 1.

Its foci are (-2 + 2, 0 + 1) = (0, 1)

and (2 + 2, 0 + 1) = (4, 3).

And its major axis is 2⋅3 = 6.

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

If the vertices of the ellipse are (±*a*, 0), (0, ±*b*)

and *a* < *b*,

then the equation of the ellipse is also*x*^{2}/*a*^{2} + *y*^{2}/*b*^{2} = 1.

## Example 3: Foci (0, -2) and (0, 2), Major Axis 8, Ellipse?

The foci are (0, 2) and (0, -2).

Then *c* = 2.

The major axis is 8.

And the *x* values of the foci are the same.

So the major axis is the vertical diameter: 2*b*.

So 2*b* = 8.

Divide both sides by 2.

Then *b* = 4.

*b* = 4*c* = 2

The major axis is 2*b*.

So *a* < *b*.

So 4^{2} - *a*^{2} = 2^{2}.

4^{2} = 16

2^{2} = 4

Move 16 to the right side.

Then -*a*^{2} = -12.

Multiply both sides by -1.

Then *a*^{2} = 12.

Instead of finding the value of *a*,

directly use *a*^{2} = 12

to write the equation of the ellipse.

*a*^{2} = 12*b* = 4

So the equation of the ellipse is*x*^{2}/12 + *y*^{2}/4^{2} = 1.

4^{2} = 16

So [*x*^{2}/12 + *y*^{2}/16 = 1] is the answer.

This is the graph of *x*^{2}/12 + *y*^{2}/16 = 1.

Its foci are (0, 2) and (0, -2).

And its major axis is 2⋅4 = 8.