$(a)$ Potential at infinity is zero. Potential energy of the system, $p-e$ $=$ Potential energy at infinity – Potential energy at distance, $d =0-\frac{q_{1} q_{2}}{4 \pi \epsilon_{0} d}$

Where, $\epsilon_{0}$ is the permittivity of free space

$\frac{1}{4 \pi \epsilon_{0}}=9 \times 10^{9} \,N\,m ^{2} \,C ^{-2}$

$\therefore$ Potential energy $=0-\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{0.53 \times 10^{10}}$$=-43.7 \times 10^{-19}\,J$

since $1.6 \times 10^{-19}\, J =1 \,eV$

$\therefore$ Potential energy $=-43.7 \times 10^{-19}=\frac{-43.7 \times 10^{-19}}{1.6 \times 10^{-19}}$$=-27.2\, eV$

Therefore, the potential energy of the system is $-27.2 \,eV.$

$(b)$ Kinetic energy is half of the magnitude of potential energy.

Kinetic energy $=\frac{1}{2} \times(-27.2)=13.6 \,eV$

Total energy $=13.6-27.2=13.6 \,eV$

Therefore, the minimum work required to free the electron is $13.6 \,eV.$

$(c)$ When zero of potential energy is taken, $d_{1}=1.06\, \mathring A$

$\therefore$ Potential energy of the system $=$ Potential energy at $d _{1}-$ Potential energy at $d$ $=\frac{q_{1} q_{2}}{4 \pi \epsilon_{0} d_{1}}-27.2 \,eV$

$=\frac{9 \times 10^{9} \times\left(1.6 \times 10^{-19}\right)^{2}}{1.06 \times 10^{-10}}-27.2\, eV$

$=21.73 \times 10^{-19}\, J -27.2 \,eV$

$=13.58\, eV -27.2\, eV$

$=-13.6 \,eV$