IIT-JEE 1996Question 1
Find all non-zero complex numbers $Z$ satisfying $\overline{Z}=iZ^{2}$.
Complex Numbers
Answer$Z=i,\ \dfrac{\sqrt3}{2}-\dfrac{i}{2},\ -\dfrac{\sqrt3}{2}-\dfrac{i}{2}$
Solution
Taking moduli, $|Z|=|Z|^2$ so $|Z|=1$. With $Z=e^{i\theta}$: $e^{-i\theta}=ie^{2i\theta}$, i.e. $e^{3i\theta}=-i$, so $\theta=-\frac\pi6+\frac{2k\pi}{3}$: $\theta=-\frac\pi6,\frac\pi2,\frac{7\pi}6$. Hence $Z=\frac{\sqrt3}{2}-\frac i2,\ i,\ -\frac{\sqrt3}{2}-\frac i2$.