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  4. If z1 and z2 are two non-zero complex numbers such that |z1+z2| =|z1|+|z2|, then arg ((z1/z2)) is equal to

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

Complex Numbers and Quadratic Equations

Solution:

Given $ \left|z_{1}+z_{2}\right|=\left|z_{1}\right|+\left|z_{2}\right|$
We know that $\left|z_{1}+z_{2}\right|\le\left|z_{1}\right|+\left|z_{2}\right|$
Equality holds when $z_{1}$, $z_{2}$ are collinear with the origin.
$\Rightarrow \,arg\left(\frac{z_{1}}{z_{2}}\right)=arg\, z_{1}- arg\,z_{2} =0$