# FIND THE EQUATION OF THE ELLIPSE WITH THE GIVEN INFORMATION

## About "Find the Equation of the Ellipse with the Given Information"

Find the Equation of the Ellipse with the Given Information :

Here we are going to see some practice questions on find the equation of the ellipse with the given information.

## Find the Equation of the Ellipse with the Given Information - Practice questions

Question 1 :

Find the equation of the ellipse in each of the cases given below:

(i) foci (± 3 0), e  =  1/2

Solution :

F1 (3, 0) and F2 (-3, 0) and e = 1/2

From the given information, we know that the given ellipse is symmetric about x axis.

Midpoint of foci  =  Center of the ellipse

Center  =  (3 + (-3))/2, (0 + 0)/2  =  C (0, 0)

Distance between foci  =  √(x2 - x1)2 + (y2 - y1)2

=  √(3 + 3)2 + (0 - 0)2

=  √62 + (0 - 0)2

2ae  =  6

ae  =  3

a(1/2)  =  3

a  =  6

b2  =  a2 (1 - e2)

b2  =  62 (1 - (1/2)2)

b2  =  36 (3/4)

b2  =  27

(x2/a2) + (y2/b2)  =  1

(x2/36) + (y2/27)  =  1

(ii) foci (0, ± 4) and end points of major axis are (0, ± 5).

Solution :

F1 (0, 4) and F2 (0, -4)

From the given foci, we know that the ellipse is symmetric about y-axis.

Distance between foci  =  √(x2 - x1)2 + (y2 - y1)2

2ae  =  √(0 - 0)2 + (4 + 4)2

2ae  =  8

ae  =  4

Distance between end points of major axis

=  √(0 - 0)2 + (5 + 5)2

2a  =  10

a  =  5

5e  =  4

e  =  4/5

b=  a2(1 - e2)

b2  =  52 (1 - (4/5)2)

b2  =  52 (9/25)

b2  =  9

Hence the required equation of ellipse is

(x2/25) + (y2/9)  =  1

(iii)  length of latus rectum 8, eccentricity = 3/and major axis on x -axis.

Solution :

Length of latus rectum  =  8

2b2/a  =  8

b2  =  4a

e  =  3/5

Since the major axis is on x-axis, the ellipse is symmetric about x-axis.

b2  =  a2 (1 - e2)

4a  =  a2 (1 - (3/5)2)

4  =  a (16/25)

a  =  25/4

b2  =  4(25/4)

b2  =  25

(x2/(16/625)) + (y2/25)  =  1

(625x2/16) + (y2/25)  =  1

(iv) length of latus rectum 4 , distance between foci 42 and major axis as y - axis.

Solution :

length of latus rectum = 4

2b2/a  =  4

b2  =  2a -------(1)

Distance between foci  =  4

2ae  =  4

ae  =  2

b =  a2 (1 - e2)

b =  a2 - (ae)2

b =  a2 - (2√2)2

b =  a2 - 8 -------(2)

2a  =  a2 - 8

a2 - 2a  - 8  =  0

(a - 4) (a + 2)  =  0

a  =  4 and a = -2

If a = 4, then b2  =  2(4)  =  8

If a = -2, then b2  =  2(-2)  =  -4 (Not admissible)

(x2/8) + (y2/16)  =  1  After having gone through the stuff given above, we hope that the students would have understood, "Find the Equation of the Ellipse with the Given Information".

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