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