Explanation

(d)
Given curve is $xy = c^{2}$
On differentiating w.r.t. $x$, we get
$\frac{dy}{dx} = \frac{- c^{2}}{x^{2}}$
Length of subnormal $= y\frac{dy}{dx}$
$= \frac{y \times ( - c)^{2}}{x^{2}} = \frac{- yc^{2}}{\left( \frac{c^{2}}{y} \right)^{2}}\ \ \left\lbrack from\ Eq.(i),\ x = \frac{c^{2}}{y} \right\rbrack$
$= \frac{- yc^{2}y^{2}}{c^{4}} = \frac{- y^{3}}{c^{2}}$
$\therefore$ Subnormal varies as $y^{3}$