Question

Let z and w be two nonzero complex numbers such that $\left|z\right|=\left|w\right|$ and $arg\left(z\right)+arg\left(w\right)=\pi$.

Then prove that $z=-\overline{w}$.

Answer 1

Let $arg\left(w\right)=\theta$. Then $arg\left(z\right)=\pi -\theta .$

Therefore, $w=\left|w\right|(cos+isin\theta )$
and $z=\left|z\right|\left(cos\right(\pi -\theta )+isin(\pi -\theta \left)\right)=\left|w\right|(-cos\theta +isin\theta )$ [$\therefore \left|z\right|=\left|w\right|$]
$=-\left|w\right|(cos\theta -isin\theta )=-\overline{w}$.
Ans: 1