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If $z_{1}$ and $z_{2}$ are two non-zero complex numbers such that $\left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$ then $\arg z_{1}-\arg z_{2}$ is equal to

1) $\frac{\pi}{2}$
2) $-\pi$
3) 0
4) $-\frac{\pi}{2}$
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(3)

Explanation

Let $Z_{1}=r_{1} e^{i r}, Z_{2}=r_{2} e^{i q_{2}}$
$\left|Z_{1}+Z_{2}\right|^{2}=\left(\left|Z_{1}+Z_{2}\right|\right)^{2}$
$\Rightarrow r_{1}^{2}+r_{2}^{2}+2 r_{1} r_{2}\left(Q_{1}-Q_{2}\right)=r_{1}^{2}+r_{2}^{2}+2 r_{1} r_{2}$
$\Rightarrow \cos \left(Q_{1}-Q_{2}\right)=1 \Rightarrow Q_{1}-Q_{2}=0$
$\therefore \arg \left(Z_{1}\right)-\arg \left(Z_{2}\right)=0$

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