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

• A

  $\frac{1}{2}$
• Correct answer
  B

  $\frac{1}{\sqrt{2}}$
• C

  $\frac{2}{3}$
• D

  $\frac{1}{3}$

Solution

Slope of $S B, m_{1}=\frac{b-0}{0-a e}=-\frac{b}{a e}$ and slope of $S^{\prime} B, m_{2}=\frac{b-0}{0-(-a e)}=\frac{b}{a e}$ since, $\angle S B S^{\prime}$ is a right angle. $\therefore$ $$m_{1} m_{2}=-1$$ $\Rightarrow \quad \frac{-b}{a e} \times \frac{b}{a e}=-1$ $\Rightarrow \quad b^{2}=a^{2} e^{2}$ $\Rightarrow \quad \frac{b^{2}}{a^{2}}=e^{2}$ $\Rightarrow \quad 1-e^{2}=e^{2}$ $\Rightarrow \quad 2 e^{2}=1$ $\Rightarrow \quad e^{2}=\frac{1}{2}$ $\Rightarrow \quad e=\pm \frac{1}{\sqrt{2}}$ $\Rightarrow \quad e=\frac{1}{\sqrt{2}}$ or $e=-\frac{1}{\sqrt{2}}$

Hence, the correct answer is option: 'B'.