# Ellipse

How to solve ellipse problems: definition, formulas, proof, examples, and their solutions.

## Definition

An ellipse is the set of points

whose sum of the distances from the foci

is constant.

(= major axis, shown below)

The sum of the blue segments' lengths is constant.

## Major Axis, Minor Axis 1

Major axis: longest diameter of the ellipse

(blue, 2*a*)

Minor axis: shortest diamter of the ellipse

(green, 2*b*)

## Formula 1

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

and if the foci are (±*c*, 0),

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

And *a*^{2} - *b*^{2} = *c*^{2}.

If *a* > *b*, the foci are on the *x*-axis.

So, if *a* > *b*, use this formula.

## Proof: Formula 1

Set [distance between (*x*, *y*) and (-*c*, 0)]

+ [distance between (*x*, *y*) and (*c*, 0)]

= 2*a*.

Move one of the radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms

so that only the remaining radical term

is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (*a*^{2} - *c*^{2}) by factoring.

Then put *b*^{2}

into *a*^{2} - *c*^{2}.

Divide both sides by *a*^{2}*b*^{2}.

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

And the relationship between *a*, *b*, and *c* is*a*^{2} - *b*^{2} = *c*^{2}.

## Example 1

To find *a*, *b*, and *c*,

change the equation into the standard form.*x*^{2}/5^{2} + *y*^{2}/4^{2} = 1

Then *a* = 5 and *b* = 4.

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

So (major axis) = 2⋅5 = 10.

And (minor axis) = 2⋅4 = 8.

*a* > *b*

So use *a*^{2} - *b*^{2} = *c*^{2}

to find *c*.*c*^{2} = *a*^{2} - *b*^{2}

= 5^{2} - 4^{2}*c* = ±3

So the foci are (±3, 0).

This is the graph of the ellipse

you've solved above.

## Example 2

Roughly draw the given conditions.

Foci: (0, 1), (4, 1)

The distance between the foci is 2|*c*|.

The foci are located horizontally.

So the horizontal axis is the major axis.

So 2*a* = 8.

The distance between the foci (0, 1), (4, 1)

is 2|*c*|.

So 2|*c*| = 4 - 0 = 4.*c* = ±2

The foci are not (±2, 0).

So the ellipse is under a translation.

Translation of a point

So use *c* = 2

to find the changes of *x* and *y*:

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

So the ellipse is under the translation

(*x*, *y*) → (*x* + 2, *y* + 1).

The major axis is 2*a* = 8.

So *a* = 4.

*a* > *b*

So *a*^{2} - *b*^{2} = *c*^{2}.

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

Write the equation of the ellipse

using the above conditions.

(*x* - 2)^{2}/4^{2} + (*y* - 1)^{2}/12 = 1

Translation of a function

See how (*x* - 2)^{2}/4^{2} + (*y* - 1)^{2}/12 = 1 looks like.

The graph is determined by

its foci (+2 + 2, +1), (-2 + 2, +1)

and its major axis 2⋅4.

## Major Axis, Minor Axis 2

If the vertical axis is greater than the horizontal axis,

then the vertical axis is the major axis

(green, 2*b*)

and the horizontal axis is the minor axis.

(blue, 2*a*)

## Formula 2

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

and if the foci are (0, ±*c*),

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

And *b*^{2} - *a*^{2} = *c*^{2}.

If *a* < *b*, the foci are on the *y*-axis.

So, if *a* < *b*, use this formula.

## Proof: Formula 2

Set [distance between (*x*, *y*) and (0, -*c*)]

+ [distance between (*x*, *y*) and (0, *c*, 0)]

= 2*b*.

Move one of the radical term to the right side.

Square both sides.

Square of a sum

Cancel the gray terms.

Then arrange the terms

so that only the remaining radical term

is on the left side.

Divide both sides by 4.

Square both sides.

Square of a sum

Cancel the gray terms.

Make (*b*^{2} - *c*^{2}) by factoring.

Then put *a*^{2}

into *b*^{2} - *c*^{2}.

Divide both sides by *a*^{2}*b*^{2}.

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

And the relationship between *a*, *b*, and *c* is*b*^{2} - *a*^{2} = *c*^{2}.

## Example 3

To find *a*, *b*, and *c*,

change the equation into the standard form.*x*^{2}/2^{2} + *y*^{2}/3^{2} = 1

Then *a* = 2 and *b* = 3.

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

So (major axis) = 2⋅3 = 6.

And (minor axis) = 2⋅2 = 4.

*a* < *b*

So use *b*^{2} - *a*^{2} = *c*^{2}

to find *c*.*c*^{2} = *b*^{2} - *a*^{2}

= 3^{2} - 2^{2}*c* = ±√5

So the foci are (0, ±√5).

This is the graph of the ellipse

you've solved above.