Cengage Multicorrect JEE 1

Multiplying both sides by $|z|$ , we get $|z-\bar{z}|z||=|z|+|z|^2$ .

Using similarity between vectors and complex numbers: $|\vec{a}+\vec{b}|≤|\vec{a}|+|\vec{b}|$ (equality when both have same directions), there exists the inequality $|a+b|\leq|a|+|b|$ (equality, when both have same arguments); or $|a-b|\leq|a|+|b|$ (equality, when the argument of one differs from the other by $\pi$ ).

Therefore $arg(z)=arg(\bar{z}|z|)±\pi$ . Using properties of arguments, we get $\frac{z}{\bar{z}|z|}=-k$ for some positive real k.

Let $z=re^{i\theta}$ , so that on simplifying we get $e^{2i\theta}=-kr<0$ (and real). Therfore, $e^{i\theta}$ is a real multiple of $i$ .