Question

If $\left|z\right|=1$ and $w=\frac{z-1}{z+1}(z\ne -1)$ then Re($w$) is

• A. $0$
• B. $\frac{-1}{{|z+1|}^2}$
• C. $\frac{1}{{|z+1|}^2}$
• D. $\frac{\sqrt{2}}{{|z+1|}^2}$

Answer 1

The correct option is A $0$

$\left|z\right|=1\&w=\frac{z-1}{z+1}$
$Re\left(w\right)=\frac{w+\overline{w}}{2}=\frac{1}{2}[\left(\frac{z-1}{z+1}\right)+\left(\frac{\overline{z}-1}{\overline{z}+1}\right)]=\frac{{\left|z\right|}^2+z-\overline{z}-1+{\left|z\right|}^2-z+\overline{z}-1}{2{|z+1|}^2}$
$\Rightarrow Re\left(w\right)=0$ ...{ $\because \left|z\right|=1$}
Hence, option 'A' is correct.