Question

The number of complex number z such that $|z|=1$ and $|z/\overline{z}+\overline{z}/z|=1$ is $\left(arg\right(z\left)ϵ\right[0,2\pi \left)\right)$

• A. 4
• B. 6
• C. 8
• D. More than 8

Answer 1

The correct option is C 8

Let $z=cosx+isinx,xϵ[0,2\pi )$. Then,
$1=|\frac{z}{\overline{z}}+\frac{\overline{z}}{z}|$
$=\frac{|z^2+{\overline{z}}^2|}{|z{|}^2}$
$=|cos2x+isin2x+cos2x-isin2x|$
$=2|cos2x|$
$\Rightarrow cos2x=\pm 1/2$
Now, $cos2x=1/2$
$\Rightarrow x_1=\frac{\pi }{6},x_2=\frac{5\pi }{6},x_3=\frac{7\pi }{6},x_4=\frac{11\pi }{6}$
$cos2x=-\frac{1}{2}$
$\Rightarrow x_5=\frac{\pi }{3},x_6=\frac{2\pi }{3},x_7=\frac{4\pi }{3},x_8=\frac{5\pi }{3}$