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 .........

Answer 1

Since, angle $FBF'$ is right angled.
$\therefore $ (Slope of $FB) = -1$

$\Rightarrow \bigg ( \frac {0-b}{ae-0} \bigg ) .\bigg (\frac {0-b}{-ae-0} \bigg ) = 1 $
$\Rightarrow \frac{b^2}{-a^2e^2}= - 1\Rightarrow \, {b^2} = {a^2e^2}$
$\Rightarrow {a^2} (1- {e^2}) = {a^2}{b^2}$
$\Rightarrow {e^2} = 1/2 \Rightarrow e = 1 \sqrt2 $