Question

If $\frac{x^2}{2}+\frac{y^2}{1}=1$ is an ellipse with foci $S_1$ and $S_2$ .Rectangle $S_{1}P{S}_{2}Q$ is completed where $P$ and $Q$ are points on ellipse then which among the following options are correct

• A. Number of such pair $P,Q$ is four
• B. Area of rectangle $S_{1}P{S}_{2}Q$ is $2$ sq. unit
• C. There will be infinite such pairs $P,Q$
• D. Rectangle $S_{1}P{S}_{2}Q$ is a square

Answer 1

Answer: (B), (D)

Rectangle $S_{1}P{S}_{2}Q$ is a square


From rectangle $S_{1}P{S}_{2}Q$
$(S_1{S}_2{)}^2=(S_{1}P{)}^2+(S_{2}P{)}^2$
$\Rightarrow (2ae{)}^2=a^2[(1-e\cos \theta {)}^2+(1+e\cos \theta {)}^2]$
$\Rightarrow 2e^2=1+e^2{\cos }^2\theta \cdots \left(1\right)$

$\because b^2=a^2(1-e^2)\Rightarrow e^2=\frac{1}{2}$

$\therefore$ From equation $\left(1\right)$, We have
$\cos \theta =0\Rightarrow \theta =\frac{\pi }{2},\frac{3\pi }{2}$
So, no of possible such pairs for $P,Q$ is $2$
$\because S_{1}P=S_{2}P=a$ .
So, rectangle $S_{1}P{S}_{2}Q$ is square having area
$\left(S_{1}P\right)\left(S_{2}P\right)=a^2=2$ sq.units