Ellipse: Equation

How to use the ellipse equation to find the major axis, the minor axis, and the foci (and vics versa): definition, formula, 8 examples, and their solutions.

Definition

Definition

An ellipse is the set of points
whose sum of the distances from the foci
is constant.

PF + PF' = (constant)

Formula: x2/a2 + y2/b2 = 1

Equation

This is the graph of the ellipse
x2/a2 + y2/b2 = 1
(a > b).

The denominator of x2, a2,
is greater than
the denominator of y2, b2.

Then this is a horizontal ellipse.

To show that
the ellipse is a horizontal ellipse,
we write x2 term first.

Major Axis

The major axis is the longest diameter.

The major axis is 2a.

Minor Axis

The minor axis is the shortest diameter.

The minor axis is 2b.

Foci

For the horizontal ellipse
x2/a2 + y2/b2 = 1,
the foci are (c, 0) and (-c, 0).

a, b, and c satisfy
a2 - b2 = c2.

Example 1: Major Axis

Example

Solution

25 = 52
16 = 42

x2/52 + y2/42 = 1

5 is greater than 4.

Then the major axis is
2⋅5.

2⋅5 = 10

So 10 is the answer.

Example 2: Minor Axis

Example

Solution

You just found that
the given ellipse is
x2/52 + y2/42 = 1.

x2/52 + y2/42 = 1

4 is the less than 5.

Then the minor axis is
2⋅4.

2⋅4 = 8

So 8 is the answer.

Example 3: Foci

Example

Solution

You found that
the given ellipse is
x2/52 + y2/42 = 1.

x2/52 + y2/42 = 1

a = 5
b = 4

Then
c2 = 52 - 42.

52 = 25
-42 = -16

25 - 16 = 9

c2 = 9

Then c = √9.

Think the sign of the c plus.

9 = 32

32 = 3

Square Root

c = 3

See x2/52 + y2/42 = 1.

The denominator of x2, 52,
is greater than
the denominator of y2, 42.

Then the ellipse is a horizontal ellipse.

So the foci are
(3, 0) and (-3, 0).

So
(3, 0), (-3, 0)
is the answer.

Example 4: Equation

Example

Solution

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

So draw a horizontal ellipse like this.
And draw the foci (4, 0) and (-4, 0).

Then c = 4.

The major axis is this horizontal diameter.
It's 10.

So 2a = 10.

Divide both sides by 2.

Then a = 5.

a = 5

The foci are (4, 0) and (-4, 0).
So c = 4.

Then
52 - b2 = 42.

52 = 25

42 = 16

Move 25 to the right side.

Then -b2 = -9.

Divide both sides by -1.

Then b2 = 9.

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

The ellipse is a horizontal ellipse.
So write the x2 term first.

a = 5
b2 = 9

Then the ellipse is
x2/52 + y2/9 = 1.

52 = 25

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

Formula: y2/a2 + x2/b2 = 1

Equation

This is the graph of the ellipse
y2/a2 + x2/b2 = 1
(a > b).

The denominator of y2, a2,
is greater than
the denominator of x2, b2.

Then this is a vertical ellipse.

To show that
the ellipse is a vertical ellipse,
we write y2 term first.

Major Axis

The major axis is 2a.

Minor Axis

The minor axis is 2b.

Foci

For the vertical ellipse
y2/a2 + x2/b2 = 1,
the foci are (0, c) and (0, -c).

a, b, and c satisfy
a2 - b2 = c2.

Example 5: Major Axis

Example

Solution

To make the right side 1,
divide both sides by 36.

The denominator of y2, 9,
is greater than
the denominator of x2, 4.

Then write the y2 term, y2/9, first.

9 = 32
4 = 22

y2/32 + x2/22 = 1

3 is greater than 2.

Then the major axis is
2⋅3.

2⋅3 = 6

So 6 is the answer.

Example 6: Minor Axis

Example

Solution

You just found that
the given ellipse is
y2/32 + x2/22 = 1.

y2/32 + x2/22 = 1

2 is less than 3.

Then the minor axis is
2⋅2.

2⋅2 = 4

So 4 is the answer.

Example 7: Foci

Example

Solution

You just found that
the given ellipse is
y2/32 + x2/22 = 1.

y2/32 + x2/22 = 1

a = 3
b = 2

Then
c2 = 32 - 22.

32 = 9
-22 = -4

9 - 4 = 5

c2 = 5

Then c = √5.

Think the sign of the c plus.

c = √5

See y2/32 + x2/22 = 1.

The denominator of y2, 32,
is greater than
the denominator of x2, 22.

Then the ellipse is a vertical ellipse.

So the foci are
(0, √5) and (0, -√5).

So
(0, √5), (0, -√5)
is the answer.

Example 8: Equation

Example

Solution

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

So draw a vertical ellipse like this.
And draw the foci (0, 2) and (0, -2).

Then c = 2.

The major axis is this vertical diameter.
It's 8.

So 2a = 8.

Divide both sides by 2.

Then a = 4.

a = 4

The foci are (0, 2) and (0, -2).
So c = 2.

Then
42 - b2 = 22.

42 = 16

22 = 4

Move 16 to the right side.

Then -b2 = -12.

Multiply both sides by -1.

Then b2 = 12.

Instead of finding b,
use b2 = 12
to write the ellipse equation.

The ellipse is a vertical ellipse.
So write the y2 term first.

a = 4
b2 = 12

Then the ellipse is
y2/42 + x2/12 = 1.

42 = 16

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