Question

If z is a complex number such that $arg\left(z\right(1+\overline{z}\left)\right)+arg\left(\frac{|z{|}^2}{z-|z^2|}\right)=0$ then

• A. $arg\overline{z}=-\frac{\pi }{2}$
• B. $argz=\frac{\pi }{4}$
• C. $|\overline{z}|<1$
• D. $ln{\left(\frac{1}{|z|}\right)}^{\prime }ϵ(-\infty ,\infty )$

Answer 1

The correct option is C $|\overline{z}|<1$

$arg\left(z\right(1+\overline{z}\left)\right)+arg\left(\frac{z\overline{z}}{z(1-\overline{z})}\right)=0$

$\Rightarrow argz+arg(1+\overline{z})+arg\overline{z}-arg(1-\overline{z})=0$

$\Rightarrow arg\left(\frac{1+\overline{z}}{1-\overline{z}}\right)=0$

$\Rightarrow \frac{1+\overline{z}}{1-\overline{z}}=k>0$

$\Rightarrow \overline{z}=\frac{k-1}{k+1}$

$\Rightarrow |\overline{z}|<1$