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# If $z_1\:\:and\:\:z_2$ are two complex numbers such that $|z_1+z_2|=|z_1|+|z_2|$, then $argz_1-argz_2=?$

$\begin{array}{1 1}(A) \; \pi\\(B)\;\large\frac{\pi}{2}\\(C)\;-\large\frac{\pi}{2}\\(D)\; 0 \end{array}$

Can you answer this question?

Let $z_1=r_1(cos\theta_1+isin\theta_1)\:\:and\:\:z_2=r_2(cos\theta_2+isin\theta2)$
$\Rightarrow\:z_1+z_2=(r_1cos\theta_1+r_2cos\theta_2)+i(r_1sin\theta_1+sin\theta_2)$
$|z_1|=r_1,\:|z_2|=r_2,\:argz_1=\theta_1,\:argz_2=\theta_2\:\:and$
$|z_1+z_2|=\sqrt {(r_1cos\theta_1+r_2cos\theta_2)^2+(r_1sin\theta_1+r_2sin\theta_2)^2}$
$=\sqrt{r_1^2+r_2^2+2r_1r_2cos(\theta_1-\theta_2)}$
If $|z_1+z_2|=|z_1|+|z_2|,$ then $cos(\theta_1-\theta_2)=1$
$\Rightarrow\:\theta_1-\theta_2=0$
$\Rightarrow\:argz_1-argz_2=0$
answered Jul 8, 2013