Explanation

(a) Current given by cell
$I = \frac{E}{R + r}$
Power delivered in first case
$P_{1} = I^{2}R_{1} = \left( \frac{E}{R_{1} + r} \right)^{2}R_{1}$
Power delivered in second case
$P_{2} = I^{2}R_{2} = \left( \frac{E}{R_{2} + r} \right)^{2}R_{2}$
Power delivered is same in the both the cases.
$\left( \frac{E}{R_{1} + r} \right)^{2}R_{1} = \left( \frac{E}{R_{2} + r} \right)^{2}R_{2}$
$\frac{R_{1}}{\left( R_{1} + r \right)^{2}} = \frac{R_{2}}{\left( R_{2} + r \right)^{2}}$
$R_{1}\left( R_{2}^{2} + r^{2} + 2R_{2}r \right) = R_{2}\left( R_{1}^{2} + r^{2} + 2R_{1}r \right)$
$R_{1}R_{2}^{2} + R_{1}r^{2} + 2R_{1}R_{2}r = R_{2}R_{1}^{2} + R_{2}r^{2} + 2R_{1}R_{2}r$
$R_{1}R_{2}^{2} - R_{2}R_{1}^{2} = R_{2}r^{2} - R_{1}r^{2}$
$R_{1}R_{2}\left( R_{2} - R_{1} \right) = r^{2}\left( R_{2} - R_{1} \right)$
$r = \sqrt{R_{1}R_{2}}$