A) \[z=1+i\]
B) \[\left| z \right|=2\]
C) \[z=1-i\]
D) \[\left| z \right|=1\]
Correct Answer: D
Solution :
\[\frac{z-1}{z+1}=\frac{x+iy-1}{x+iy+1}\] \[\frac{z-1}{z+1}=\frac{{{x}^{2}}+{{y}^{2}}-1+2iy}{{{x}^{2}}+{{y}^{2}}+2x+1}\] \[\Rightarrow \,\,\,\operatorname{Re}\left( \frac{z-1}{z+1} \right)=\frac{{{x}^{2}}+{{y}^{2}}-1}{{{x}^{2}}+{{y}^{2}}+2x+1}=0\] \[\Rightarrow \,\,\,\,\,{{x}^{2}}+{{y}^{2}}-1=0\] \[\Rightarrow \,\,\,\,\,{{x}^{2}}+{{y}^{2}}=1\] Also, \[z\bar{z}={{x}^{2}}+{{y}^{2}}=1\] \[z\bar{z}={{\left| z \right|}^{2}}\] \[\Rightarrow \,\,\,\,{{\left| z \right|}^{2}}=1\] \[\Rightarrow \,\,\,\,\left| z \right|=1\]
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