Question

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

• A. $\frac{\sqrt{3}}{2}-\frac{i}{2},-\frac{\sqrt{3}}{2}-\frac{i}{2},-i$
• B. $\frac{\sqrt{3}}{2}+\frac{i}{2},-\frac{\sqrt{3}}{2}+\frac{i}{2},i$
• C. $\frac{\sqrt{3}}{2}-\frac{i}{2},-\frac{\sqrt{3}}{2}-\frac{i}{2},i$
• D. $\frac{\sqrt{3}}{2}+\frac{i}{2},-\frac{\sqrt{3}}{2}+\frac{i}{2},-i$

Answer 1

Answer: (C)

$\frac{\sqrt{3}}{2}-\frac{i}{2},-\frac{\sqrt{3}}{2}-\frac{i}{2},i$

Let $z=e^{i\theta }$
Hence
$e^{-i\theta }=e^{i(\frac{\pi }{2}+2\theta )}$
$1=e^{(\frac{\pi }{2}+3\theta )i}$
Or
$e^{i2k\pi }=e^{(\frac{\pi }{2}+3\theta )i}$
Hence
$2k\pi =(\frac{\pi }{2}+3\theta )$
If $k=0$ , we get
$\theta =\frac{-\pi }{6}$ ...($arg\left(z_1\right)$)
If $k=1$ we get
$2\pi =\frac{\pi }{2}+3\theta$
$3\theta =\frac{3\pi }{2}$
$\theta =\frac{\pi }{2}$ ...($arg\left(z_2\right)$)
Hence answer is option C.