Question

$B$ is an extremity of the minor axis of the ellipse whose foci are $S$ and $S^{\prime }.$ If $\angle SB{S}^{\prime }$ is a right angle, then eccentricity of the ellipse is

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

Answer 1

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


Coordinates of $S=(ae,0)$ and $S^{\prime }=(-ae,0)$
whereas coordinate of $B(0,b)$
From above image $OS=O{S}^{\prime }$
$\angle OSB=45°$
From $\tan 45°=\frac{b}{ae}\Rightarrow b=ae$
$\Rightarrow b^2=a^2{e}^2$
$\Rightarrow a^2(1-e^2)=a^2{e}^2\Rightarrow e=\frac{1}{\sqrt{2}}$