Question

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 5√3 ), then the length of its latus rectum is:
(A) 6 (B) 5
(C) 8 (D) 10

Answer 1

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Answer 2

In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is 10 and one of the foci is at (0, 5√3 ), then the length of its latus rectum is given by:

Given,

The difference of the lengths of major axis and minor axis = 10

⇒ 2b - 2a = 10

⇒ b - a = 5 ............ 1

Given,

One of the foci is at (0, 5√3 )

⇒2be = 2 × 5√3 = 10√3

⇒ be = 5√3

⇒b²e² = 25 × 3 = 75

⇒ b² - a² = 75

⇒ (b + a) (b - a) = 75

⇒ (b + a) × 5 = 75

⇒ b + a  = 15 ...........2

solving 1 and 2 we get,

a = 5

b = 10

Length of latus rectum is given by,

l = 2a²/b

= (2 × 5²) / 10

= 50 / 10

= 5

Therefore, the length of latus rectum is 5.

Option (B) is correct.