JEE Math Practice Question

Let $S$ be the set of all complex numbers $Z$ satisfying $|z-2+i| \geq \sqrt{5}$. If the complex number $Z_{0}$ is such that $\frac{1}{\left|Z_{0}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|z-1|}: z \in S\right\}$, then the principal argument of $\frac{4-z_{0}-\overline{z_{0}}}{Z_{0}-\overline{z_{0}}+2 i}$ is

1. A.$-\frac{\pi}{2}$
2. B.$\frac{\pi}{4}$
3. C.$\frac{\pi}{2}$
4. D.$\frac{3 \pi}{4}$