Question

If $z_1$ and $z_2$ are conjugate to each other, and $\arg (-z_1{z}_2)=k\pi$, then $k=$

Answer 1

$z_1$ and $z_2$ are conjugate to each other, i.e., $z_2=\overline{z_1}$.
$\therefore \arg (-z_1{z}_2)=k\pi \Rightarrow k\pi =\arg (-z_1\overline{z_1})\Rightarrow k\pi =\arg (-|z_1{|}^2)\Rightarrow k\pi =\arg \left(\text{a negative real number}\right)\Rightarrow k\pi =\pi \therefore k=1$