Explanation

(b)
Normal at $\left( ae,\ \frac{b^{2}}{a} \right)$ of ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is
$\frac{x - ae}{\frac{ae}{a^{2}}} = \frac{y - \frac{b^{2}}{a}}{\left( \frac{\frac{b^{2}}{a}}{b^{2}} \right)}$
Since, it passes through $(0,\ - b)$, then
$\frac{0 - ae}{\frac{ae}{a^{2}}} = \frac{- b - \frac{b^{2}}{a}}{\frac{1}{a}}$
$\Rightarrow {- a}^{2} = - a\left( b + \frac{b^{2}}{a} \right)$
$\Rightarrow a^{2} = ab + b^{2}$
$\Rightarrow a^{2} = ab + a^{2} - a^{2}e^{2}\ \ \left( \because b^{2} = a^{2} - a^{2}e^{2} \right)$
$\Rightarrow b = {ae}^{2}$
$\Rightarrow b^{2} = a^{2}e^{4}$
$\Rightarrow a^{2}\left( 1 - e^{2} \right) = a^{2}e^{4}$
$\Rightarrow 1 - e^{2} = e^{4}$
$\Rightarrow e^{2}\left( e^{2} + 1 \right) = 1$