Question

Number of complex number $z$ such that $|z|=1$ and $a=\frac{z}{\overline{z}}$

$|a+\frac{1}{a}|$ is

• A. $4$
• B. $6$
• C. 0
• D. infinite

Answer 1

Answer: (D)

infinite

Let, $z=\cos x+i\sin x$
$\Rightarrow |\frac{z}{\overline{z}}+\frac{\overline{z}}{z}|=\left|\frac{z^2+(\overline{z}{)}^2}{z\overline{z}}\right|=1$
$\Rightarrow |\cos 2x+i\sin 2x+\cos 2x-i\sin 2x|=1$
$=2|\cos 2x|=1$
$\Rightarrow \cos 2x=\frac{1}{2}or\cos 2x=-\frac{1}{2}$
$\Rightarrow x=n\pi \pm \frac{\pi }{6},m\pi \pm \frac{\pi }{3},m,n\in I$