A) \[|z+\bar{z}|=\frac{1}{2}\]
B) \[z+\bar{z}=1\]
C) \[|z+\bar{z}|=1\]
D) None of these
Correct Answer: D
Solution :
If \[\left| z-1 \right|>\left| z+1 \right|,\]then max \[\{|z-1|,|z+1|\}=|z-1|\] \[\Rightarrow \] If \[|z{{|}^{2}}+1-z-\bar{z}>|z{{|}^{2}}+1+z+\bar{z}\] then \[|z|=|z-1|\] \[\Rightarrow \] If \[z+\bar{z}<0\] then \[|z{{|}^{2}}=|z{{|}^{2}}+1-z-\bar{z}\] \[\Rightarrow \] If \[z+\bar{z}<0\] then \[z+\bar{z}=1,\] which is not possible. Again If \[|z+1|>|z-1|\] then max \[\{|z-1|,|z+1|\}=|z+1|\] \[\Rightarrow \] If \[|z{{|}^{2}}+1+z+\bar{z}>\,|z{{|}^{2}}+1-z-\bar{z}\] then \[|z|\,=\,|z+1|\] \[\Rightarrow \] If \[z+\bar{z}>0\] then \[|z{{|}^{2}}=|z{{|}^{2}}+1+z+\bar{z}\] \[\Rightarrow \] If \[z+\bar{z}>0\]then \[z+\bar{z}=-1\] Not possible again. Therefore the given result cannot hold.
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