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 \[\theta \,=\,135{}^\circ \] |
\[p\,({{z}_{0}})=\left\{ \left( 2+\sqrt{5}\left( \frac{-\,1}{\sqrt{2}} \right) \right),\left( -1+\sqrt{5}\left( \frac{1}{\sqrt{2}} \right) \right) \right\}\] |
\[\Rightarrow \] \[{{z}_{0}}\,=\,\left( 2-\sqrt{\frac{5}{2}},-1+\sqrt{\frac{5}{2}} \right)\] |
\[\Rightarrow \] \[\text{arg}\left( \frac{4-({{z}_{0}}+{{{\bar{z}}}_{0}})}{({{z}_{0}}-\bar{z})+2i} \right)\] |
\[\Rightarrow \] \[\text{arg}\left[ \frac{4-\left( 2\left\{ 2-\sqrt{\frac{5}{2}} \right\} \right)}{2i+2\left( -1+\sqrt{\frac{5}{2}} \right)i} \right]\] |
\[\Rightarrow \] \[\text{arg}\left( \frac{\sqrt{10}}{i\sqrt{10}} \right)\] \[\Rightarrow \] \[\text{arg}\,\left( \frac{1}{i} \right)\] |
\[\Rightarrow \] \[\text{arg}(-i)\,=\,\frac{-\pi }{2}\] |
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