Question

The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to ______.

विकल्प

• 1

• 2

• 3

• 4

Answer 1

The ratio of the area of the ellipse and the area enclosed by the locus of mid-point of PS where P is any point on the ellipse and S is the focus of the ellipse, is equal to 4.

Explanation:

Let `x^2/a^2 + y^2/b^2` = 1

be an ellipse

Area = A₁ = π ab

Let (h, k) be the mid-point of PS

⇒ 2h = a cos θ + ae and 2k =  b sin θ

Eliminating θ, we get

`(x - (ae)/2)^2/(a/2)^2 + y^2/(b/2)^2` = 1

A₂ = `π a/2 b/2 = (πab)/4`

⇒ A₁ : A₂ = 4 : 1