Mathematics - Complex Number Question with Solution | TestHub

Let $z_1=\frac{w-{\bar{w}}_z}{1-z}$, be purely real. $$\begin{array}{rlrl}\Rightarrow & z_1& ={\bar{z}}_1& \\ & \therefore & \frac{w-\bar{w}z}{1-z}& =\frac{\bar{w}-w\bar{z}}{1-\bar{z}}\\ \Rightarrow & & w-w\bar{z}-\bar{w}z+\bar{w}z\cdot \bar{z}& =\bar{w}-z\bar{w}-w\bar{z}+wz\cdot \bar{z}\\ \Rightarrow & & (w-\bar{w})+(\bar{w}-w)|z{|}^2& =0\\ \Rightarrow & & (w-\bar{w})\left(1-|z{|}^2\right)& =0\\ \Rightarrow & & |z{|}^2& =1\\ \Rightarrow & & |z|& =1\text{ and }z\ne 1.\end{array}$$ $$\text{ [as, }w-\bar{w}\ne 0\text{, since }\beta \ne 0\text{ ] }$$