Question

An ellipse has $OB$ as a semi-minor axis. $F$ and $F^{\prime }$ are its foci and the angle $FB{F}^{\prime }$ is a right angle. Then, the eccentricity of the ellipse is .........

Answer 1

Since, angle $FB{F}^{\prime }$ is right angled.
$\therefore$ (Slope of $FB)=-1$

$\Rightarrow \left(\frac{0-b}{ae-0}\right).\left(\frac{0-b}{-ae-0}\right)=1$
$\Rightarrow \frac{b^2}{-a^2{e}^2}=-1\Rightarrow b^2=a^2{e}^2$
$\Rightarrow a^2(1-e^2)=a^2{b}^2$
$\Rightarrow e^2=1/2\Rightarrow e=1\sqrt{2}$