Question

If the angle between the lines joining the end points of minor axis of an ellipse with its foci is $\frac{\pi }{2}$, then the eccentricity of the ellipse is

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

Answer 1

Answer: (B)

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

Since $\angle FB{F}^{\prime }=\frac{\pi }{2}$ (Given)

$\therefore \angle FBC=\angle F^{\prime }BC=\frac{\pi }{4}$

$CB=CF\Rightarrow b=ae\Rightarrow b^2=a^2{e}^2$

$\Rightarrow a^2(1-e^2)=a^2{e}^2$

$\Rightarrow 1-e^2=e^2\Rightarrow 2e^2=1\Rightarrow e=\frac{1}{\sqrt{2}}$