Question

If $|z_1|=|z_2|$ and $\arg \left(\frac{z_1}{z_2}\right)=\pi$, then value of $z_1+z_2$ is

Answer 1

We have,
$|z_1|=|z_2|\Rightarrow \frac{|z_1|}{|z_2|}=1\Rightarrow \left|\frac{z_1}{z_2}\right|=1$
and $\arg \left(\frac{z_1}{z_2}\right)=\pi$

$\therefore \frac{z_1}{z_2}=\left|\frac{z_1}{z_2}\right|(\cos \pi +i\sin \pi )\Rightarrow \frac{z_1}{z_2}=1\cdot (-1+0)\Rightarrow z_1=-z_2\Rightarrow z_1+z_2=0$