A) \[|z+\bar{z}|=\frac{1}{2}\]
B) \[z+\bar{z}=1\]
C) \[|z+\bar{z}|=1\]
D) None of these
Correct Answer: C
Solution :
| \[\left| z \right|=\left| z-1 \right|\Rightarrow {{\left| z \right|}^{2}}={{\left| z-1 \right|}^{2}}\] |
| \[\Rightarrow \,\,\,z\,\bar{z}=\left( z-1 \right)\,\left( \overline{z}-1 \right)\] |
| \[\Rightarrow \,\,\,z\overline{z}=z\overline{z}-z-\overline{z}+1\] |
| \[\therefore \,\,\,z+\overline{z}=1\] |
| \[\left| z \right|=\left| z+1 \right|\Rightarrow z+\overline{z}=-1\] |
| \[\therefore \,\,\left| z+\overline{z} \right|=1\] |
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