Question

Number of complex numbers $z$ 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

Answer: (A)

$4$

We have, $|\frac{z^2+{\overline{z}}^2}{z\overline{z}}|=1$

Let $z=x+iy$, and we know $z\overline{z}=1$

$\therefore |z^2+{\overline{z}}^2|=1\Rightarrow |x^2-y^2|=\frac{1}{2}$

and it's given that, $x^2+y^2=1$

$\therefore 2x^2=\frac{3}{2}\Rightarrow x=\pm \frac{\sqrt{3}}{2}$

And $y=\pm \frac{1}{2}$

Therefore, Number of complex numbers $z$ are $4$.