Ellipse

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.

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. Minor axis: shortest diamter of the ellipse.

Major axis: longest diameter of the ellipse
(blue, 2a)

Minor axis: shortest diamter of the ellipse
(green, 2b)

Formula 1

If the vertices of the ellipse are (+-a, 0), (0, +-b), and if the foci are (+-c, 0), then the ellipse is (x^2)/(a^2) + (y^2)/(b^2) = 1. And a^2 - b^2 = c^2.

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
x2/a2 + y2/b2 = 1.

And a2 - b2 = c2.

If a > b, the foci are on the x-axis.
So, if a > b, use this formula.

Proof: Formula 1

Ellipse: Proof of the Formula 1

Set [distance between (x, y) and (-c, 0)]
+ [distance between (x, y) and (c, 0)]
= 2a.

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 (a2 - c2) by factoring.

Then put b2
into a2 - c2.

Divide both sides by a2b2.

Then the equaition of the ellipse is
x2/a2 + y2/b2 = 1.

And the relationship between a, b, and c is
a2 - b2 = c2.

Example 1

The equation of an ellipse is given below. (x^2)/25 + (y^2)/16 = 1 1. Find the lengths of the major and minor axes. 2. Find the coordinates of the foci.

To find a, b, and c,
change the equation into the standard form.

x2/52 + y2/42 = 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 a2 - b2 = c2
to find c.

c2 = a2 - b2
= 52 - 42
c = ±3

So the foci are (±3, 0).

This is the graph of the ellipse
you've solved above.

Example 2

Write an equation of the ellipse described below. foci: (0, 1), (4, 1), major axis: 8

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 2a = 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 2a = 8.
So a = 4.

a > b

So a2 - b2 = c2.
42 - b2 = 22

b2 = 12

Write the equation of the ellipse
using the above conditions.

(x - 2)2/42 + (y - 1)2/12 = 1

Translation of a function

See how (x - 2)2/42 + (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 and the horizontal axis is the minor axis.

If the vertical axis is greater than the horizontal axis,

then the vertical axis is the major axis
(green, 2b)

and the horizontal axis is the minor axis.
(blue, 2a)

Formula 2

If the vertices of the ellipse are (+-a, 0), (0, +-b), and if the foci are (0, +-c), then the ellipse is (x^2)/(a^2) + (y^2)/(b^2) = 1. And b^2 - a^2 = c^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
x2/a2 + y2/b2 = 1.

And b2 - a2 = c2.

If a < b, the foci are on the y-axis.
So, if a < b, use this formula.

Proof: Formula 2

Ellipse: Proof of the Formula 2

Set [distance between (x, y) and (0, -c)]
+ [distance between (x, y) and (0, c, 0)]
= 2b.

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 (b2 - c2) by factoring.

Then put a2
into b2 - c2.

Divide both sides by a2b2.

Then the equaition of the ellipse is
x2/a2 + y2/b2 = 1.

And the relationship between a, b, and c is
b2 - a2 = c2.

Example 3

The equation of an ellipse is given below. 9x^2 + 4y^2 = 36 1. Find the lengths of the major and minor axes. 2. Find the coordinates of the foci.

To find a, b, and c,
change the equation into the standard form.

x2/22 + y2/32 = 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 b2 - a2 = c2
to find c.

c2 = b2 - a2
= 32 - 22
c = ±√5

So the foci are (0, ±√5).

This is the graph of the ellipse
you've solved above.