Ellipse: Formula

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.

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

then the equation of the ellipse is
x2/a2 + y2/b2 = 1.

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

Write an equation of the ellipse whose foci and major axis are below. Foci: (4, 0), (-4, 0). Major axis: 10

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: 2a.

So 2a = 10.

Divide both sides by 2.

Then a = 5.

a = 5
c = 4

The major axis is 2a.
So a > b.

So 52 - b2 = 42.

52 = 25

42 = 16

Move 25 to the right side.

Then -b2 = -9.

Multiply both sides by -1.

Then b2 = 9.

Instead of finding the value of b,
directly use b2 = 9
to write the equation of the ellipse.

a = 5
b2 = 9

So the equation of the ellipse is
x2/52 + y2/9 = 1.

52 = 25

So [x2/25 + y2/9 = 1] is the answer.

This is the graph of x2/25 + y2/9 = 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?

Write an equation of the ellipse whose foci and major axis are below. Foci: (0, 1), (4, 1). Major axis: 6

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 2c.

So 2c = 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: 2a.

So 2a = 6.

Divide both sides by 2.

Then a = 3.

a = 3
c = 2

The major axis is 2a.
So a > b.

So 32 - b2 = 22.

Ellipse - Foci (a > b)

32 = 9

22 = 4

Move 9 to the right side.

Then -b2 = -5.

Multiply both sides by -1.

Then b2 = 5.

Instead of finding the value of b,
directly use b2 = 5
to write the equation of the ellipse.

a = 3
b2 = 5

The ellipse is under the translation
(x, y) → (x + 2, y + 1).

So the equation of the ellipse is
(x - 2)2/32 + (y - 1)2/5 = 1.

Translation of a function

32 = 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.

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

then the equation of the ellipse is also
x2/a2 + y2/b2 = 1.

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

Write an equation of the ellipse whose foci and major axis are below. Foci: (0, 2), (0, -2). Major axis: 8

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: 2b.

So 2b = 8.

Divide both sides by 2.

Then b = 4.

b = 4
c = 2

The major axis is 2b.
So a < b.

So 42 - a2 = 22.

42 = 16

22 = 4

Move 16 to the right side.

Then -a2 = -12.

Multiply both sides by -1.

Then a2 = 12.

Instead of finding the value of a,
directly use a2 = 12
to write the equation of the ellipse.

a2 = 12
b = 4

So the equation of the ellipse is
x2/12 + y2/42 = 1.

42 = 16

So [x2/12 + y2/16 = 1] is the answer.

This is the graph of x2/12 + y2/16 = 1.

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

And its major axis is 2⋅4 = 8.