Explanation

(b) Here, $I = I_{1} + I_{2}$ ….(i)
and $\frac{I_{1}}{I_{2}} = \frac{1}{2} = 2$
$\therefore I = I_{1} + I_{2} = 2I_{2} + I_{2}$
$\therefore I_{2} = \frac{I}{3}\ and\ I_{1}\frac{2I}{3}$
$\therefore P = I^{2} \times 3 = 3I^{2}$
${\ P}_{1} = I_{1}^{2} \times 1 = \left( \ \frac{2I}{3} \right)^{2} \times 1 = \frac{4I^{2}}{9}$
$P_{2} = I_{2}^{2} \times 2 = \frac{I^{2}}{9} \times 2 = \frac{2}{9}I^{2}$
$P_{3} = I^{2} \times 3 = 3I^{2}$
$\therefore P_{1}:P_{2}:P_{3} = \frac{4}{9}:\frac{2}{9} = 4\ :2\ :27$