Mathematics - Ellipse Question with Solution | TestHub

In standard ellipse  $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Semi-major axis $OA=a$

Semi-minor axis $OB=b$

Focus $F_1\equiv \left(-ae,0\right)$

$F_2\equiv \left(ae,0\right)$

$\angle F_{1}B{F}_2=90°$

$O{F}_1=O{F}_2\Rightarrow \angle F_{1}BO=\angle F_{2}BO=45°$

$\Rightarrow \frac{ae}{b}=tan45°=1$ 

$\Rightarrow b=ae$

Also, $b^2=a^2\left(1-e^2\right)$

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

$\Rightarrow 2e^2=1$

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