Question

If $z_1$ and $z_2$ are two non-zero complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then $arg{z}_1-arg{z}_2$ is equal to

• A. $-\pi$
• B. $\pi /2$
• C. $-\pi /2$
• D. 0

Answer 1

The correct option is D 0

Given $z_1+z_2|=|z_1|+|z_2|$
$\Rightarrow |z_1{|}^2+|z_2{|}^2+2Re\left(z_1\overline{z_2}\right)=|z_1{|}^2+|z_2{|}^2+2|z_1||z_2|$
$\Rightarrow Re\left(z_1\overline{z_2}\right)=|z_1||z_2|$
$\Rightarrow Re\left(z_1\overline{z_2}\right)=|z_1\overline{z_2}|$
(or $Re\left(\overline{z_1}{z}_2\right)=|\overline{z_1}{z}_2|)$
$\Rightarrow z_1\overline{z_2}=R^{+}\Rightarrow Arg\left(z_1\overline{z_2}\right)=Arg\left(\overline{z_1}{z}_2\right)=0$
$\Rightarrow Arg{z}_1+Arg\overline{z_2}=Arg\overline{z_1}+Arg{z}_2=0$
$\Rightarrow Arg{z}_1-Arg{z}_2=-Arg{z}_1+Arg{z}_2=0$