A) \[|z-(5-i)|=5\]
B) \[|z-(5-i)|=\sqrt{5}\]
C) \[|z-(5+i)|=5\]
D) \[|z-(5+i)|=\sqrt{5}\]
Correct Answer: B
Solution :
\[\because \] \[\arg \left( \frac{z-{{z}_{1}}}{{{z}_{2}}-z} \right)=\frac{\pi }{2}\] \[\Rightarrow \] \[\operatorname{Re}\left( \frac{z-{{z}_{1}}}{{{z}_{2}}-z} \right)=0\] \[\Rightarrow \] \[\frac{z-{{z}_{1}}}{{{z}_{2}}-z}+\frac{\overline{z}-{{\overline{z}}_{1}}}{{{\overline{z}}_{2}}-\overline{z}}=0\] \[\Rightarrow \]\[(z-{{z}_{1}})({{\overline{z}}_{2}}-\overline{z})+({{z}_{2}}+-z)(\overline{z}-{{\overline{z}}_{1}})=0\] \[\Rightarrow \]\[z({{\overline{z}}_{1}}+{{\overline{z}}_{2}})+\overline{z}({{z}_{1}}+{{z}_{2}})2z\overline{z}\] \[-({{z}_{1}}{{\overline{z}}_{2}}+{{z}_{2}}\overline{{{z}_{1}}})=0\] \[\Rightarrow \]\[z\overline{z}-(5+i)z+(5-i)\overline{z}+21=0\] \[\therefore \] \[|z-(5-i)|=\sqrt{5}\]
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