As an electron makes a transition from an excited state to the ground state of a hydrogen – like atom/ion

Explain Rutherford's argument for scattered $\alpha $ -particles. 

Particles used in the Rutherford's scattering experiment to deduce the structure of atoms

In an atom, two electrons move around the nucleus in circular orbits of radii $R$ and $4R.$ The ratio of the time taken by them to complete one revolution is : (neglect electric interaction)

The following diagram indicates the energy levels of a certain atom when the system moves from $4E$ level to $E$. A photon of wavelength $\lambda _1$ is emitted. The wavelength of photon produced during it's transition from $\frac{7}{3}E$ level to $E$ is $\lambda_2$. The ratio $\frac{{{\lambda _1}}}{{{\lambda _2}}}$ will be