Question

If z is a complex number such that $z^2=2(\overline{z}{)}^2$, then

• A. z is purely real
• B. z is purely imaginary
• C. either z is purely real or purely imaginary
• D. none of these

Answer 1

The correct option is C either z is purely real or purely imaginary

Let $z=x+iy\Rightarrow \overline{z}=x-iy$
Given $z^2=2(\overline{z}{)}^2\Rightarrow x^2-y^2+2xyi=2(x^2-y^2)-4xyi$
Comparing coefficients we get $x^2=y^2$ and $xy=0$
$\Rightarrow x=0$ or $y=0\therefore z$ is either purely real or purely imaginary.