Question

An ellipse has OB as a semi minor axis. F and F’ are its foci, and the angle FBF’ is a right angle. Then the eccentricity of the ellipse is

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

Answer 1

The correct option is B

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

FBF' is ${90}^0$

Applying Pythagoras' theorem in $△FB{F}^{\prime }$:

$(F^{\prime }B{)}^2+(FB{)}^2=(F^{\prime }F{)}^2$

$\Rightarrow b^2+a^2{e}^2+a^2{e}^2+b^2=4a^2{e}^2$

$\Rightarrow 2b^2+2a^2{e}^2=4a^2{e}^2$

$\Rightarrow e^2=\frac{b^2}{a^2}$

We know that $e^2=1-\frac{b^2}{a^2}$

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