Explanation

(d) Let $R_{1}\ and\ R_{2}$ be the resistances of the coils
$H = \frac{V^{2}t_{1}}{R_{1}}\ and\ \ H = \frac{V^{2}t_{2}}{R_{2}}$
$\Rightarrow \frac{t_{1}}{R_{1}} = \frac{t_{2}}{R_{2}},\ ie,\frac{R_{2}}{R_{1}} = \frac{t_{2}}{t_{1}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(i)$
Now in parallel
$R^{'} = \frac{R_{1}R_{2}}{R_{1} + R_{2}} = R_{1}$
$\therefore H = \frac{V^{2}t}{R^{'}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(ii)$
Now, $\frac{V^{2}t}{R^{'}} = \frac{V^{2}t_{1}}{R_{1}}$
$\Rightarrow \frac{t \times \left( R_{1} + \ R_{2} \right)}{R_{1}R_{2}} = \frac{t_{1}}{R_{1}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(iii)$
On using Eqs. (i) and (iii), we get
$t = \frac{t_{1}t_{2}}{t_{1} + t_{2}}$