Question

If $z$ is a complex number satisfying $|z|=1$, then the range of $\arg \left(\frac{1}{1-z}\right)$ is

• A. $(0,\pi )$
• B. $[0,\pi ]$
• C. $[0,\frac{\pi }{2}]$
• D. $(-\frac{\pi }{2},\frac{\pi }{2})$

Answer 1

The correct option is D $(-\frac{\pi }{2},\frac{\pi }{2})$

Let $u=\frac{1}{1-z}$
$\Rightarrow z=1-\frac{1}{u}\Rightarrow |z|=|1-\frac{1}{u}|\Rightarrow |1-\frac{1}{u}|=1\Rightarrow |u-1|=|u|$
Locus of $u$ is the perpendicular bisector of the line segment joining points $(0,0)$ and $(1,0)$
$Re\left(u\right)=\frac{1}{2}$
Therefore,
$\arg \left(u\right)\in (-\frac{\pi }{2},\frac{\pi }{2})$