# FIND THE EQUATION OF ELLIPSE FROM FOCI AND LENGTH OF MAJOR OR MINOR AXIS

In ellipse vertices, foci and center they lie in the same line and on the major axis.

• Midpoint of foci is center.
• Midpoint of vertices is center.
• Length of major axis is 2a
• Length of minor axis is 2b.

Problem 1 :

Foci: (±5, 0); major axis of length 12

Solution:

Foci are F(5, 0) and F(-5, 0). By observing the given foci, the ellipse is symmetric about x-axis.

Length of major axis = 12

2a = 12

a = 6

Midpoint of foci = center

Here the foci are on the x-axis, so the major axis is along the x-axis.

So, the equation of the ellipse is

2a = 12

a = 6

a2 = 36

c = 5

b2 = a2 - c2

b2 = 62 - 52

b2 = 36 - 25

b2 = 11

Hence the required equation of ellipse is Problem 2 :

Foci: (±2, 0); major axis of length 8

Solution:

Given the major axis is 8 and foci are (±2, 0).

Here the foci are on the x-axis, so the major axis is along the x-axis.

So, the equation of the ellipse is

2a = 8

a = 4

a2 = 16

c = 2

b2 = a2 - c2

b2 = 42 - 22

b2 = 16 - 4

b2 = 12

Hence the required equation of ellipse is Problem 3 :

Foci: (0, 0), (4, 0); major axis of length 8

Solution:

The midpoint between the foci is the center

The distance between the foci is equal to 2c

The major axis length is equal to 2a

2a = 8

a = 4

b2 = a2 - c2

= 42 - 22

= 16 - 4

b2 = 12

The standard equation of an ellipse with a horizontal major axis is Problem 4 :

Foci: (0, 0), (0, 8); major axis of length 16

Solution:

The midpoint between the foci is the center

The distance between the foci is equal to 2c

The major axis length is equal to 2a

2a = 16

a = 8

b2 = a2 - c2

b2 = 82 - 42

b2 = 64 - 16

b2 = 48

By observing foci, since x-coordinates are same. The ellipse is symmetric about y-axis. Problem 5 :

Vertices: (0, 4), (4, 4); minor axis of length 2

Solution:

The center of the ellipse

By observing center and foci, the ellipse is symmetric about x-axis.

Length of minor axis = 2

2b = 2

b = 1

Length of the major axis

a2 = 16

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