Question

If $|z|=1$ , then $\arg \left(\sqrt{\frac{(1+z)}{(1-z)}}\right)$ will be

• A. $\frac{\pi }{6}\text{or}\frac{5\pi }{6}$
• B. $\frac{\pi }{3}\text{or}\frac{2\pi }{3}$
• C. $\frac{\pi }{4}\text{or}\frac{3\pi }{4}$
• D. $\frac{\pi }{2},\pi$

Answer 1

The correct option is C $\frac{\pi }{4}\text{or}\frac{3\pi }{4}$


Since $|z|=1,z$ lies on a unit circle having centre at the origin
Form the above diagram
$\arg \left(\frac{1+z}{1-z}\right)=\frac{\pi }{2}\text{or}\frac{3\pi }{2}$
$\Rightarrow \frac{1+z}{1-z}=k{e}^{i\pi /2}\text{or}k{e}^{i3\pi /2}$
where $k$ is a real parameter and its value depends upon the position of $z$.
$\arg \left(\sqrt{\frac{(1+z)}{(1-z)}}\right)=\frac{\pi }{4}\text{or}\frac{3\pi }{4}$