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

Problem 1 :

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

Solution:

Foci are F₁ (5, 0) and F₂ (-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

Equation of the ellipse