question_answer

Let S be the set of all complex numbers z satisfying  \[\left| z\,-\,2+i \right|\,\le \,\sqrt{5}.\] If the complex number \[{{z}_{0}}\] is such that \[\frac{1}{\left| {{z}_{0}}\,-\,1 \right|}\] is the maximum of the set \[\left\{ \frac{1}{\left| z-1 \right|}:z\,\in \,\text{S} \right\},\] then the principal argument  of \[\frac{4\,\,-\,\,{{z}_{0}}-\,\bar{z}}{{{z}_{0}}\,-\,{{{\bar{z}}}_{0}}+2i}\] is:

A) \[\frac{\pi }{2}\]                        

B) \[\frac{3\pi }{4}\]

C) \[\frac{\pi }{4}\]                        

D) \[-\,\frac{\pi }{2}\]

Correct Answer: D

Solution :

\[\left| z\,-\,2+\,i \right|\,\,\ge \,\sqrt{5}\]for max of \[\frac{1}{\left| {{z}_{0}}\,-\,1 \right|}\]

\[\Rightarrow \]\[\min |{{z}_{0}}-1|\]

\[\Rightarrow \]\[{{\text{m}}_{\text{CA}}}\,=\,\text{tan}\,\theta \,=\frac{1}{-\,1}\,=\,-1\]

Now use parametric coordinate \[\theta \,=\,135{}^\circ \]

\[p\,({{z}_{0}})=\left\{ \left( 2+\sqrt{5}\left( \frac{-\,1}{\sqrt{2}} \right) \right),\left( -1+\sqrt{5}\left( \frac{1}{\sqrt{2}} \right) \right) \right\}\]

\[\Rightarrow \]   \[{{z}_{0}}\,=\,\left( 2-\sqrt{\frac{5}{2}},-1+\sqrt{\frac{5}{2}} \right)\]

\[\Rightarrow \]   \[\text{arg}\left( \frac{4-({{z}_{0}}+{{{\bar{z}}}_{0}})}{({{z}_{0}}-\bar{z})+2i} \right)\]

\[\Rightarrow \]   \[\text{arg}\left[ \frac{4-\left( 2\left\{ 2-\sqrt{\frac{5}{2}} \right\} \right)}{2i+2\left( -1+\sqrt{\frac{5}{2}} \right)i} \right]\]

\[\Rightarrow \]   \[\text{arg}\left( \frac{\sqrt{10}}{i\sqrt{10}} \right)\] \[\Rightarrow \] \[\text{arg}\,\left( \frac{1}{i} \right)\]

\[\Rightarrow \]   \[\text{arg}(-i)\,=\,\frac{-\pi }{2}\]