Mathematics - Ellipse Question with Solution | TestHub

Let eq. of ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $\mathrm{S}$ is $(-\mathrm{ae},0),\mathrm{T}$ is $(\mathrm{ae},0)$ and $\mathrm{B}$ is $(0,\mathrm{b})$
$\Rightarrow \mathrm{SB}=\sqrt{(0+\mathrm{ae}{)}^2+{\mathrm{b}}^2}$ Also ${\mathrm{SB}}^2={\mathrm{ST}}^2\Rightarrow 4{\mathrm{a}}^2{\mathrm{e}}^2={\mathrm{a}}^2{\mathrm{e}}^2+{\mathrm{b}}^2$ $\Rightarrow 3a^2{e}^2=a^2\left(1-e^2\right)=a^2-a^2{e}^2$ $\Rightarrow 4a^2{e}^2=a^2\Rightarrow e^2=\frac{1}{4}\Rightarrow e=\frac{1}{2}$