Explanation

(c)
Equation of chord joining points $P(a\cos\alpha,\ b\sin\alpha)$ and $Q(a\cos\beta,\ b\sin\beta)$ is
$\frac{x}{a}\cos{\left( \frac{\alpha + \beta}{2} \right) + \frac{y}{b}\sin{\left( \frac{\alpha + \beta}{2} \right) = \cos\left( \frac{\alpha - \beta}{2} \right)}}$
Now, $\beta = \alpha + 90{^\circ}$
$\frac{x}{a}\cos{\left( \frac{2\alpha + 90{^\circ}}{2} \right) + \frac{y}{b}\sin{\left( \frac{2\alpha + 90{^\circ}}{2} \right) = \frac{1}{\sqrt{2}}}}$
now, compare it with $lx + my = - n$, we get
$\frac{\cos\left( \frac{2\alpha + 90{^\circ}}{2} \right)}{al} = \frac{\sin\left( \frac{2\alpha + 90{^\circ}}{2} \right)}{bm} = - \frac{1}{\sqrt{2}n}$
$\because\cos^{2}\theta + \sin^{2}\theta = 1$
$\Rightarrow a^{2}l^{2} + b^{2}m^{2} = {2n}^{2}$