First slide

Ellipse

Question

$The ratio of the area enclosed by the locus of the midpoint of P S and area of the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 is$
$(P be any point on the ellipse and S be iti focus)$

Moderate

A

$\frac{1}{2}$
B

$\frac{1}{3}$
C

$\frac{1}{5}$
D

$\frac{1}{4}$

Solution

$Mid point of PS is (h, k)$
$\begin{array}{l} ∴ h = \frac{a \cos θ + a e}{2} \Rightarrow \cos θ = \frac{2 h - a e}{a} \\ and k = \frac{b \sin θ}{2} \end{array}$
$Eliminating θ, we get locus as$
$\begin{array}{l} \frac{(2 h - a e)^{2}}{a^{2}} + \frac{4 k^{2}}{b^{2}} = 1 \\ \Rightarrow \frac{{(x - \frac{a e}{2})}^{2}}{\frac{a^{2}}{4}} + \frac{y^{2}}{\frac{b^{2}}{4}} = 1 \end{array}$
$which is ellipse having enclosed area$
$\begin{array}{l} ∴ Area = π \frac{a}{2} \cdot \frac{b}{2} = \frac{π a b}{4} \\ Hence, required ratio = \frac{1}{4} \end{array}$

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