# Ellipse

See how to solve an ellipse

(major/minor axis, foci, equation).

5 examples and their solutions.

## Definition

PF + PF' = (constant)

PF + PF' = (constant, major axis).

F, F': foci

## Ellipse: x^{2}a^{2} + y^{2}b^{2} = 1 (a > b)

### Equation

x

a

Major Axis: 2a

Minor Axis: 2b

Foci: (c, 0), (-c, 0)

The major axis is the longest diameter: 2a.^{2}a^{2}+ y^{2}b^{2}= 1 (a > b)a

^{2}- b^{2}= c^{2}Major Axis: 2a

Minor Axis: 2b

Foci: (c, 0), (-c, 0)

The minor axis is the shortest diameter: 2b.

a > b

Then the foci are located horizontally.

By using a

^{2}- b

^{2}= c

^{2},

you can find the foci (±c, 0).

### Example

x

1. Major axis?

2. Minor axis?

3. Foci?

Solution ^{2}25 + y^{2}16 = 11. Major axis?

2. Minor axis?

3. Foci?

x

x

1. (major axis) = 2⋅5

= 10

2. (minor axis) = 2⋅4

= 8

3. c

= 25 - 16

= 9

c = ±3

Foci: (3, 0), (-3, 0)

^{2}25 + y^{2}16 = 1x

^{2}5^{2}+ y^{2}4^{2}= 11. (major axis) = 2⋅5

= 10

2. (minor axis) = 2⋅4

= 8

3. c

^{2}= 5^{2}- 4^{2}= 25 - 16

= 9

c = ±3

Foci: (3, 0), (-3, 0)

Close

### Example

Foci: (4, 0), (-4, 0)

Major axis: 10

Equation of the ellipse?

Solution Major axis: 10

Equation of the ellipse?

c = 4

2a = 10

a = 5

5

^{2}- b

^{2}= 4

^{2}

25 - b

^{2}= 16

-b

^{2}= 16 - 25

-b

^{2}= -9

b

^{2}= 9

x

^{2}5

^{2}+ y

^{2}9 = 1 - [2]

x

^{2}25 + y

^{2}9 = 1

[1]

Draw the ellipse

and the foci (4, 0), (-4, 0).

Foci are located horizontally.

So the major axis, 10,

is the horizontal diameter.

So 2a = 10.

and the foci (4, 0), (-4, 0).

Foci are located horizontally.

So the major axis, 10,

is the horizontal diameter.

So 2a = 10.

[2]

a = 5

b

So the equation of the ellipse is

x

b

^{2}= 9So the equation of the ellipse is

x

^{2}/5^{2}+ y^{2}/9 = 1.Close

### Example

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

Major axis: 6

Equation of the ellipse?

Solution Major axis: 6

Equation of the ellipse?

2c = 0 + 4

2c = 4

c = 2

(2, 0) → (4, 1) = (2 + 2, 0 + 1)

(x, y) → (x + 2, y + 1) - [2]

2a = 6

a = 3

3

^{2}- b

^{2}= 2

^{2}

9 - b

^{2}= 4

-b

^{2}= 9 - 4

-b

^{2}= -5

b

^{2}= 5

(x - 2)

^{2}3

^{2}+ (y - 1)

^{2}5 = 1 - [3]

(x - 2)

^{2}9 + (y - 1)

^{2}5 = 1

[1]

Draw the ellipse

and the foci (0, 1), (4, 1).

Foci are located horizontally.

So the major axis, 6,

is the horizontal diameter.

So 2a = 6.

and the foci (0, 1), (4, 1).

Foci are located horizontally.

So the major axis, 6,

is the horizontal diameter.

So 2a = 6.

[2]

c = 2

So the right focus should be (2, 0).

But the right focus is (4, 1).

Then there's a translation

(2, 0) → (4, 1).

(4, 1) = (2 + 2, 0 + 1)

So the translation is

(x, y) → (x + 2, y + 1).

So the right focus should be (2, 0).

But the right focus is (4, 1).

Then there's a translation

(2, 0) → (4, 1).

(4, 1) = (2 + 2, 0 + 1)

So the translation is

(x, y) → (x + 2, y + 1).

[3]

a = 3

b

(x, y) → (x + 2, y + 1)

So the equation of the ellipse is

(x - 2)

b

^{2}= 5(x, y) → (x + 2, y + 1)

So the equation of the ellipse is

(x - 2)

^{2}/3^{2}+ (y - 1)^{2}/5 = 1.Close

## Ellipse: x^{2}a^{2} + y^{2}b^{2} = 1 (a < b)

### Equation

x

b

Major Axis: 2b

Minor Axis: 2a

Foci: (0, c), (0, -c)

^{2}a^{2}+ y^{2}b^{2}= 1 (a < b)b

^{2}- a^{2}= c^{2}Major Axis: 2b

Minor Axis: 2a

Foci: (0, c), (0, -c)

### Example

9x

1. Major axis?

2. Minor axis?

3. Foci?

Solution ^{2}+ 4y^{2}= 361. Major axis?

2. Minor axis?

3. Foci?

9x

x

x

1. (major axis) = 2⋅3

= 6

2. (minor axis) = 2⋅2

= 4

3. c

= 9 - 4

= 5

c = ±√5

Foci: (0, √5), (0, -√5)

^{2}+ 4y^{2}= 36x

^{2}4 + y^{2}9 = 1 - [1]x

^{2}2^{2}+ y^{2}3^{2}= 11. (major axis) = 2⋅3

= 6

2. (minor axis) = 2⋅2

= 4

3. c

^{2}= 3^{2}- 2^{2}= 9 - 4

= 5

c = ±√5

Foci: (0, √5), (0, -√5)

[1]

÷36 both sides.

Close

### Example

Foci: (0, 2), (0, -2)

Major axis: 8

Equation of the ellipse?

Solution Major axis: 8

Equation of the ellipse?

c = 2

2b = 8

b = 4

4

^{2}- a

^{2}= 2

^{2}

16 - a

^{2}= 4

-a

^{2}= 4 - 16

-a

^{2}= -12

a

^{2}= 12

x

^{2}12 + y

^{2}4

^{2}= 1 - [2]

x

^{2}12 + y

^{2}16 = 1

[1]

Draw the ellipse

and the foci (0, 2), (0, -2).

Foci are located vertically.

So the major axis, 8,

is the vertical diameter.

So 2b = 8.

and the foci (0, 2), (0, -2).

Foci are located vertically.

So the major axis, 8,

is the vertical diameter.

So 2b = 8.

[2]

a

b = 4

So the equation of the ellipse is

x

^{2}= 12b = 4

So the equation of the ellipse is

x

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