Question

If $z_1, z_2$ and $z_3, z_4$ are two pairs of conjugate complex numbers, then $\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right)$ equals

• A. $0, \pi$
• B. $\pi,-\pi$
• C. $\frac{\pi}{2}, \frac{3 \pi}{2}$
• D. $0,2 \pi$

Answer 1

Answer: (D)

$\arg \left(\frac{z_1}{z_4}\right)+\arg \left(\frac{z_2}{z_3}\right) $
$=\arg \left(\frac{z_1 z_2}{z_4 z_3}\right)+2 k \pi=\arg \left(\frac{\left|z_1\right|^2}{\left|z_3\right|^2}\right)+2 k \pi=2 k \pi$
where $k=0$ or 1
$\left[\because z_1=z_3=i\right.$, gives answer $2 \pi$ and $z_1=z_3=1$, gives answer 0]