Question

If the angle between the lines joining the foci to an extremity of minor axis of an ellipse is ${90}^{\circ }$, its eccentricity is:

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

Answer 1

The correct option is D $\frac{1}{\sqrt{2}}$

It is given that, triangle BSS' is a right angled triagled at $B$
$\therefore B{S}^{\prime 2}+B{S}^2=S{S}^{\prime 2}$
$\Rightarrow (b^2+a^2{e}^2)+(b^2+a^2{e}^2)=(2ae{)}^2$
$\Rightarrow b^2=a^2{e}^2\Rightarrow \left(1\right)$
Also we know $b^2=a^2(1-e^2)$
$\Rightarrow e^2=1-e^2$
$\Rightarrow e=\sqrt{\frac{1}{2}}$