FIND THE EQUATION OF THE ELLIPSE WITH THE GIVEN INFORMATION

Question :

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

b²  =  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 = 3/5 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 4√2 and major axis as y - axis.

Solution :

length of latus rectum = 4

2b²/a  =  4

b²  =  2a -------(1) 

Distance between foci  =  4√2 

2ae  =  4√2 

ae  =  2√2 

b²  =  a² (1 - e²)

b²  =  a² - (ae)²

b²  =  a² - (2√2)²

b²  =  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

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