Question

Find all non zero complex numbers $z$ satisfying $\overline{z}=i{z}^2$.

• A. $(0,1),(\pm \sqrt{3/4},1/2)$
• B. $(0,1),(\pm 1/4,-1/2)$
• C. $(0,1),(\pm \sqrt{3/2},-1/2)$
• D. $(0,1),(\pm 1/4,1/2)$

Answer 1

The correct option is C $(0,1),(\pm \sqrt{3/2},-1/2)$

We know that

$z\overline{z}=|z{|}^2$

Given:

$\overline{z}=i{z}^2$ ...(1)

Taking mod on both sides, we get

$|z|=|z{|}^2$

$\Rightarrow |z|=0$ or $|z|=1$

It is given the $z$ is a non-zero number, therefore $|z|=1$

Now,

Multiply (1) both sides with with $z$, we get

$z\overline{z}=i{z}^3$

$\Rightarrow |z{|}^2=i{z}^3$

$\Rightarrow z^3=-i$

$\Rightarrow z^3-i^3=0$

$\Rightarrow (z-i)(z^2+iz-1)=0$
Hence
$z=i$ and $z^2+iz-1=0$

$z^2+iz-1=0$ Gives
$z=\frac{-i\pm \sqrt{-1+4}}{2}$
$z=\frac{-i\pm \sqrt{3}}{2}$

So $z$ can be $(0,1)$ or $(\pm \frac{\sqrt{3}}{2},-\frac{1}{2})$

Hence, option C.