S be the set of all complex numbers z satisfying $|\mathrm{z}-2+\mathrm{i}|\ge \sqrt{5}$. If the complex number Z₀ is such that $\frac{1}{\left|{\mathrm{z}}_{\mathrm{o}}-1\right|}$ is the maximum of the set $\left\{\frac{1}{|\mathrm{z}-1|}:\mathrm{Z}\in \mathrm{S}\right\}$, then the principal argument of $\frac{4-{\mathrm{z}}_{\mathrm{o}}-\overline{{\mathrm{z}}_{\mathrm{o}}}}{{\mathrm{z}}_{\mathrm{o}}-\overline{{\mathrm{z}}_{\mathrm{o}}}+2\mathrm{i}}$ is

$\frac{1}{|\mathrm{z}-1|},\mathrm{z}\in \mathrm{S}$ attains maximum when $|\mathrm{z}-1|$ attains minimum $|\mathrm{z}-2+\mathrm{i}|\ge \sqrt{5}$ represents a circle with centre (2, -1) and radius $\sqrt{5}\frac{4-{\mathrm{Z}}_{\mathrm{o}}-{\overline{\mathrm{Z}}}_{\mathrm{o}}}{{\mathrm{Z}}_{\mathrm{o}}-{\overline{\mathrm{Z}}}_{\mathrm{o}}+2\mathrm{i}}=\frac{4-2\mathrm{Re}\left({\mathrm{Z}}_{\mathrm{o}}\right)}{2{\mathrm{iI}}_{\mathrm{m}}\left({\mathrm{Z}}_{\mathrm{o}}\right)+2\mathrm{i}}=\frac{2-\mathrm{Re}{\mathrm{Z}}_{\mathrm{o}}}{\mathrm{i}\left({\mathrm{I}}_{\mathrm{m}}{\mathrm{Z}}_{\mathrm{o}}+1\right)}=-\mathrm{i\lambda },\mathrm{\lambda }>\mathrm{o}$ Since 2 - Re z_o > o and I_mz₀ > 0

The principal argument of $\frac{4-{\mathrm{z}}_{\mathrm{o}}-{\overline{\mathrm{z}}}_{\mathrm{o}}}{\mathrm{i}\left({\mathrm{I}}_{\mathrm{m}}{\mathrm{z}}_{\mathrm{o}}+1\right)}\mathrm{is}-\frac{\mathrm{\pi }}{2}$