# 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: x^{2}/a^{2} + y^{2}/b^{2} = 1

### Equation

This is the graph of the ellipse

x^{2}/a^{2} + y^{2}/b^{2} = 1

(a > b).

The denominator of x^{2}, a^{2},

is greater than

the denominator of y^{2}, b^{2}.

Then this is a horizontal ellipse.

To show that

the ellipse is a horizontal ellipse,

we write x^{2} 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

x^{2}/a^{2} + y^{2}/b^{2} = 1,

the foci are (c, 0) and (-c, 0).

a, b, and c satisfy

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

## Example 1: Major Axis

### Example

### Solution

25 = 5^{2}

16 = 4^{2}

x^{2}/5^{2} + y^{2}/4^{2} = 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

x^{2}/5^{2} + y^{2}/4^{2} = 1.

x^{2}/5^{2} + y^{2}/4^{2} = 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

x^{2}/5^{2} + y^{2}/4^{2} = 1.

x^{2}/5^{2} + y^{2}/4^{2} = 1

a = 5

b = 4

Then

c^{2} = 5^{2} - 4^{2}.

5^{2} = 25

-4^{2} = -16

25 - 16 = 9

c^{2} = 9

Then c = √9.

Think the sign of the c plus.

9 = 3^{2}

√3^{2} = 3

Square Root

c = 3

See x^{2}/5^{2} + y^{2}/4^{2} = 1.

The denominator of x^{2}, 5^{2},

is greater than

the denominator of y^{2}, 4^{2}.

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

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

5^{2} = 25

4^{2} = 16

Move 25 to the right side.

Then -b^{2} = -9.

Divide both sides by -1.

Then b^{2} = 9.

Instead of finding b,

use b^{2} = 9

to write the ellipse equation.

The ellipse is a horizontal ellipse.

So write the x^{2} term first.

a = 5

b^{2} = 9

Then 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.

## Formula: y^{2}/a^{2} + x^{2}/b^{2} = 1

### Equation

This is the graph of the ellipse

y^{2}/a^{2} + x^{2}/b^{2} = 1

(a > b).

The denominator of y^{2}, a^{2},

is greater than

the denominator of x^{2}, b^{2}.

Then this is a vertical ellipse.

To show that

the ellipse is a vertical ellipse,

we write y^{2} term first.

### Major Axis

The major axis is 2a.

### Minor Axis

The minor axis is 2b.

### Foci

For the vertical ellipse

y^{2}/a^{2} + x^{2}/b^{2} = 1,

the foci are (0, c) and (0, -c).

a, b, and c satisfy

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

## Example 5: Major Axis

### Example

### Solution

To make the right side 1,

divide both sides by 36.

The denominator of y^{2}, 9,

is greater than

the denominator of x^{2}, 4.

Then write the y^{2} term, y^{2}/9, first.

9 = 3^{2}

4 = 2^{2}

y^{2}/3^{2} + x^{2}/2^{2} = 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

y^{2}/3^{2} + x^{2}/2^{2} = 1.

y^{2}/3^{2} + x^{2}/2^{2} = 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

y^{2}/3^{2} + x^{2}/2^{2} = 1.

y^{2}/3^{2} + x^{2}/2^{2} = 1

a = 3

b = 2

Then

c^{2} = 3^{2} - 2^{2}.

3^{2} = 9

-2^{2} = -4

9 - 4 = 5

c^{2} = 5

Then c = √5.

Think the sign of the c plus.

c = √5

See y^{2}/3^{2} + x^{2}/2^{2} = 1.

The denominator of y^{2}, 3^{2},

is greater than

the denominator of x^{2}, 2^{2}.

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

4^{2} - b^{2} = 2^{2}.

4^{2} = 16

2^{2} = 4

Move 16 to the right side.

Then -b^{2} = -12.

Multiply both sides by -1.

Then b^{2} = 12.

Instead of finding b,

use b^{2} = 12

to write the ellipse equation.

The ellipse is a vertical ellipse.

So write the y^{2} term first.

a = 4

b^{2} = 12

Then the ellipse is

y^{2}/4^{2} + x^{2}/12 = 1.

4^{2} = 16

So

y^{2}/16 + x^{2}/12 = 1

is the answer.