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# If $z\:\:and\:\:w$ are two non zero complex numbers such that $|zw|=1$ and $Arg(z)-Arg(w)=\large\frac{\pi}{2},$ then $\overline z w$ = ?

(A) 1 (B) -1 (C) i (D) -i

Let $z=r_1(cos\theta_1+isin\theta_1)$ and
$w=r_2(cos\theta_2+isin\theta_2)$
$\Rightarrow\:|z|=r_1,\:|w|=r_2,\:arg(z)=\theta_1,\:and\:arg(w)=\theta_2$
Given:$|zw|=r_1r_2=1$ and $\theta_1-\theta_2=\large\frac{\pi}{2}$
$\overline z=r(cos\theta_1-isin\theta_1)$
$=r_1(cos(-\theta_1)+isin(-\theta_1))$
$arg(\overline z)=-\theta_1$
$\Rightarrow\:\overline z w=r_1r_2(cos(-\theta_1+\theta_2)+isin(-\theta_1+\theta_2))$