Number of non zero complex number $z$ satisfying $\bar{z}=i z^{2}$.

Question

Professor PC Thomas Classes · Answer

Let $z=x+i y$. Given $\bar{z}=i z^{2}$\ $\Rightarrow \quad x-i y=i(x+i y)^{2}=i x^{2}-2 x y-i y^{2}$ $\left[\because x^{2}-y^{2}=-yight.$ and $\left.-2 x y=xight]$\ $\Rightarrow \quad y=-\frac{1}{2}$ or $x=0$. when $x=0, y=0$ or 1 when $y=-\frac{1}{2}, x= \pm \frac{\sqrt{3}}{2}$\ Hence required non zero complex numbers are $0+i, \frac{\sqrt{3}}{2}-\frac{1}{2} i, \frac{-\sqrt{3}}{2}-\frac{1}{2} i$.\