What is the eccentricity of the ellipse if the angle between the...

Let ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$
It's foci are $\mathrm{S}_1(a e, o)$ and $\mathrm{S}_2(-a e, o)$ and the extremity of the minor axis is $\mathrm{B}(o, b)$ Then, slope of $\mathrm{BS}_2=\frac{b-o}{o-(-a e)}=\frac{b}{a e}$ and $\quad$ slope of $\mathrm{BS}_1=\frac{b-o}{o-a e}=-\frac{b}{a e}$ The product of slopes $=\frac{b}{a e} \times-\frac{b}{a e}=-1\Rightarrow \quad b^2=a^2 e^2\Rightarrow \quad e^2=\frac{b^2}{a^2} \qquad$ ....(i) We have,$e=\frac{c}{a}\Rightarrow \quad c=\sqrt{a^2-b^2}\Rightarrow \quad e^2=1-\frac{b^2}{a^2}=1-e^2 \quad[$ from $(i)]$$\begin{aligned} \Rightarrow & & 2 e^2 & =1 \\ \Rightarrow & & e^2 & =\frac{1}{2} \Rightarrow e=\frac{1}{\sqrt{2}}\end{aligned}$$$