z and w are two non-zero complex numbers such that |z|=|w| and Arg(z)+Arg(w)=.

Question

$z$ and $w$ are two non-zero complex numbers such that $|z|=|w|$ and $\operatorname{Arg}(z)+\operatorname{Arg}(w)=\pi$. Then $z$ equals

Accepted Answer

Let $|z|=|w|=r$, $z=re^{i\theta}$, $w=re^{i\varphi}$ with $\theta+\varphi=\pi$.

$$z=re^{i(\pi-\varphi)}=-re^{-i\varphi}=-\overline w.$$

Solutions · 1