Explanation

(d) For such a motion, magnetic force = centripetal force
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ qvB = \frac{mv^{2}}{r}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ qB = \frac{mv}{r}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r = \frac{mv}{qB}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{r_{p}}{r_{c}} = \frac{m_{p}v_{p}}{qB}\ \times \frac{qB}{m_{e}v_{e}}$
but $\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ K_{e} = K_{p}$ (given)
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{2}\ m_{e}v_{e}^{2} = \frac{1}{2}m_{p}v_{p}^{2}$
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{m_{p}}{m_{e}} = \left( \frac{v_{e}}{v_{p}} \right)^{2}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{r_{p}}{r_{e}} = \left( \frac{v_{e}}{v_{p}} \right)^{2}\frac{v_{p}}{v_{e}} = \frac{v_{e}}{v_{p}}$
Hence, the electron and proton will follow circular paths but with different radii.