# $z$ and $\omega$ are two non zero complex numbers such that $\mid z\mid=\mid \omega\mid$ and $arg(z)+arg(\omega)=\pi$.Then $z$ equals

$\begin{array}{1 1}(A) \omega \\ (B) \bar{\omega} \\ (C) - \omega \\(D) - \bar{\omega} \end{array}$

Let $arg(\omega)=\theta$
$\Rightarrow arg(z)=\pi-\theta$
$\Rightarrow z=r(\cos(\pi-\theta)+i\sin(\pi-\theta))$
$\Rightarrow r(-\cos \theta+i\sin \theta)$
$\Rightarrow -r(\cos \theta-i\sin \theta)$
$\Rightarrow -\overline{\omega}$
Hence (D) is the correct answer.
answered Apr 9, 2014