Question

Let $S$ be the set of all complex numbers $z$ satisfying $\left|z-2+i\right|\ge 5$. If the complex number $z_0$ is such that $\frac{1}{z_0-1}$ is the maximum of the set $\left\{\frac{1}{z-1};z\in S\right\}$

then the principle argument of $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$

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

Answer 1

The correct option is B

$-\frac{\mathrm{\pi }}{2}$

Explanation of correct answer :

Finding the argument of $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$:

Given, $\left|z-2+i\right|\ge 5$

$P$ is along $AC$ but at $\sqrt{5}$ distance from $C$.

$$\begin{gathered} CP=\sqrt{5}\frac{\overrightarrow{AC}}{\left|\overrightarrow{AC}\right|}=\sqrt{5}\left(\frac{i+j}{\sqrt{2}}\right) \\ \overrightarrow{P}=\left(2i+j\right)=-\frac{\sqrt{5}}{\sqrt{2}}\left(i-j\right)=\left(2-\frac{\sqrt{5}}{\sqrt{2}}\right)i+\left(\frac{\sqrt{5}}{\sqrt{2}}-1\right)j \end{gathered}$$

$z_0=\left(2-\sqrt{\frac{5}{2}}\right)+\left(4+\sqrt{\frac{5}{2}}\right)$

arg $\frac{4-z_0-\overline{z_0}}{z_0-\overline{z_0}+2i}$,

$$\begin{gathered} z_0+\overline{z_0}=4-\sqrt{10} \\ {z}_0-\overline{z_0}=-2i+\sqrt{10}i \end{gathered}$$

arg$\left(\frac{\sqrt{10}}{\sqrt{10}i}\right)=-\frac{\mathrm{\pi }}{2}$

Hence, the argument is $-\frac{\mathrm{\pi }}{2}$.

Therefore, the correct answer is option (B).