Question

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

• A. $\frac{1}{8}$
• B. $\frac{1}{\sqrt{3}}$
• C. $\sqrt{\frac{2}{3}}$
• D. $\sqrt{\frac{1}{2}}$

Answer 1

Answer: (D)

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

In $\Delta OSA$, $\tan \frac{\pi }{4}=\frac{ae}{b}$
$\Rightarrow 1=\frac{ae}{b}$
$\Rightarrow e^2=\frac{b^2}{a^2}$
$\because e^2=1-\frac{b^2}{a^2}$
$\Rightarrow e^2=1-e^2$
$\Rightarrow e^2=\frac{1}{2}$
$\Rightarrow e=\sqrt{\frac{1}{2}}$