If the energy of a hydrogen atom in the $n^{th}$ orbit is ${E_n}$,then the energy in the $n^{th}$ orbit of a singly ionized helium atom will be:

The following parameter is the same for all hydrogen-like atoms and ions in their ground state.

Suppose an electron in an atom experiences an attractive force towards the nucleus given by $F = k/r$,where $k$ is a constant and $r$ is the distance of the electron from the nucleus. Applying the Bohr model to this system,the radius of the $n^{th}$ orbit $r_n$ and the kinetic energy of the electron in that orbit $K_n$ are found. Which of the following relationships is correct?

The radius of the orbit of an electron in a Hydrogen-like atom is $4.5 a_0$,where $a_0$ is the Bohr radius. Its orbital angular momentum is $\frac{3h}{2\pi}$. It is given that $h$ is Planck constant and $R$ is Rydberg constant. The possible wavelength$(s)$,when the atom de-excites,is (are) :
$(A)$ $\frac{9}{32R}$ $(B)$ $\frac{9}{16R}$ $(C)$ $\frac{9}{5R}$ $(D)$ $\frac{4}{3R}$

The energy level diagram for a hydrogen-like atom is shown in the figure. The radius of its first Bohr orbit is

The magnitude of acceleration of the electron in the $n$th orbit of a hydrogen atom is $a_{H}$ and that of a singly ionised helium atom is $a_{He}$. The ratio of $a_{H} : a_{He}$ is