To solve this problem, we need to use the formula for the radius of the path of a charged particle moving in a magnetic field.$\\$The radius $r$ of the path of a charged particle in a magnetic field is given by:$\\ r = \frac{p}{qB} \\$where:$\\$• $p$ is the momentum of the particle,$\\$• $q$ is the charge of the particle,$\\$• $B$ is the magnetic field.$\\$Given that both the ionized hydrogen atom and the $\alpha$-particle have the same momentum, we can write:$\\ r_H = \frac{p}{q_H B} \\ r_\alpha = \frac{p}{q_\alpha B} \\$The charge of an ionized hydrogen atom (proton) is $q_H = e,$ and the charge of an $\alpha$-particle is $q_\alpha = 2e. \\$Now, let's find the ratio of their radii:$\\ \frac{r_H}{r_\alpha} = \frac{\frac{p}{q_H B}}{\frac{p}{q_\alpha B}} \\$Simplifying the expression:$\\ \frac{r_H}{r_\alpha} = \frac{q_\alpha}{q_H} \\$Substitute the values of the charges:$\\ \frac{r_H}{r_\alpha} = \frac{2e}{e} \\ \frac{r_H}{r_\alpha} = 2 \\$Therefore, the ratio of the radii of their paths is $2 : 1. \\$This corresponds to Option 1.