A) \[\left\{ z:\left| z \right|=1 \right\}\]
B) \[\left\{ z:z=\overline{z} \right\}\]
C) \[\left\{ z:z\ne 1 \right\}\]
D) \[\left\{ z:\left| z \right|=1,z\ne 1 \right\}\]
Correct Answer: D
Solution :
Consider the equation \[w-\overline{w}z=k(1-z),k\in R\] Clearly \[z\ne 1\]and\[\frac{w-\overline{w}z}{1-z}\]is purely real \[\therefore \]\[\frac{\overline{w-wz}}{1-z}=\frac{w-\overline{w}z}{1-z}\]\[\Rightarrow \]\[\frac{\overline{w}-w\overline{z}}{1-\overline{z}}=\frac{w-\overline{w}z}{1-z}\] \[\Rightarrow \]\[\overline{w}-\overline{w}z-w\overline{z}+wz\overline{z}=w-w\overline{z}-\overline{w}z+\overline{w}z\overline{z}\] \[\Rightarrow \]\[\overline{w}+w|z{{|}^{2}}=w+\overline{w}|z{{|}^{2}}\] \[\Rightarrow \]\[(w-\overline{w})(|z{{|}^{2}})=w-\overline{w}\] \[\Rightarrow \]\[|z{{|}^{2}}=1(\because \operatorname{Im}w\ne 0)\] \[\Rightarrow \]\[|z|=1\]and\[z\ne 1\] \[\therefore \]The required set is \[\{z:|z|=1,z\ne 1\}\]
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