Question

If the foci of an ellipse subtend a right angle at either extremity of its minor axis, then its eccentricity

• A. Must be $\frac{1}{\sqrt{2}}$
• B. can be $\frac{1}{2}$ or $\frac{\sqrt{3}}{2}$
• C. Must be $\sqrt{\frac{2}{3}}$
• D. Cannot be determined

Answer 1

The correct option is A

Must be $\frac{1}{\sqrt{2}}$

Let ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Its foci are $S_1(ae,0)$ and $S_2(-ae,0)$ and the extremity of the minor axis is $B(0,b)$

Then

The product of slopes, $m_{S_{1}B}\times m_{S_{2}B}=-1$ [Because $S_{1}B$ and $S_{2}B$ are perpendicular]

$\Rightarrow \left(\frac{b-0}{0-ae}\right)\left(\frac{b-0}{0+ae}\right)=-1$

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

$\Rightarrow 1-e^2=e^2\Rightarrow e^2=\frac{1}{2}\Rightarrow e=\frac{1}{\sqrt{2}}$