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

• A. $\frac{1}{2}$
• B. $\frac{1}{3}$
• C. $\frac{1}{5}$
• D. $\frac{1}{4}$

Answer 1

The correct option is D $\frac{1}{4}$

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,Area=\pi abLetP=(acos\theta ,bsin\theta )S=(ae,0)M(h,k)midpointofPS\Rightarrow h=\frac{ae+acos\theta }{2};k=\frac{bsin\theta }{2}\Rightarrow {\left(\frac{h-\frac{ae}{2}}{\frac{a}{2}}\right)}^2+{\left(\frac{k}{\frac{b}{2}}\right)}^2=1$
Area = $\frac{1}{4}\pi ab$