A) \[\operatorname{Re}(z)=\frac{3}{2}\]
B) \[\operatorname{Re}(z)=\frac{7}{2}\]
C) \[\operatorname{Re}(z)\in \left\{ \frac{3}{2},\frac{7}{2} \right\}\]
D) None of these
Correct Answer: C
Solution :
\[|z\,-2|=\,\,min\,\left\{ \left| z-1 \right|<\left| z-5 \right| \right\}\] i.e., \[\left| z\,-2 \right| = \left| z-1 \right|,\,\,where\,\,\left| z\,-1 \right|<\left| z\,- 5 \right|\] \[\Rightarrow \,\,\,\operatorname{Re}(z) = \frac{3}{2}\,\,\,which satisfy \left| z-5 \right|\,<\,\,\left| z-1 \right|\] Also, \[\left| z\,-2 \right|=\left| z\,-5 \right|, where\, \left| z\,-5 \right|<\left| z-1 \right|\] \[\Rightarrow \,\,\,\operatorname{Re}(z)= \frac{7}{2} \,which satisfy\,\,\left| z\,-5 \right|\,\,\left| z\,-1 \right|\]
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