Question

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

Solution

The correct option is D
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, A r e a = π a b L e t P = (a c o s θ, b s i n θ) S = (a e, 0) M (h, k) m i d p o i n t o f P S \Rightarrow h = \frac{a e + a c o s θ}{2}; k = \frac{b s i n θ}{2} \Rightarrow {(\frac{h - \frac{a e}{2}}{\frac{a}{2}})}^{2} + {(\frac{k}{\frac{b}{2}})}^{2} = 1$
Area = $\frac{1}{4} π a b$

Mathematics

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