The angle between the lines joining the foci of an ellipse to one particular extremity of the minor axis is $90^{\circ}$. The eccentricity of the ellipse is

• A

  $\frac{1}{8}$
• B

  $\frac{1}{\sqrt{3}}$
• C

  $\sqrt{\frac{2}{3}}$
• Correct answer
  D

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

Solution

$$ \begin{gathered} \text { } \frac{b}{a e}=\tan \frac{\pi}{4} \\ \qquad \begin{array}{c} a e \Rightarrow \frac{b}{a}=e \\ e^2=1-\frac{b^2}{a^2} \\ e^2=1-e^2 \end{array} \end{gathered} $$ $$ e^2=\frac{1}{2} \Rightarrow e=\frac{1}{\sqrt{2}} $$

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