Question

Find all non-zero complex number $Z$ satisfying $\overline{Z}=i{Z}^2$.

Answer 1

Let, $Z=x+iy$.

Given,

$\overline{Z}=i{Z}^2$

or, $x-iy=i(x^2-y^2+2ixy)$

Comparing the real and imaginary part both sides we get,

$x=-2yx$ and $-y=x^2-y^2$.......(1).

or, $y=-\frac{1}{2}$[ Since $Z$ is non-zero].

Using the value of $y$ in (1) we get,

$x^2=\frac{3}{4}$

or, $x=\pm \frac{\sqrt{3}}{2}$

So $Z=(\pm \frac{\sqrt{3}}{2},-\frac{1}{2})$.