Question

If $|z_1|=|z_2|$ and $arg\left(z_1/z_2\right)=\pi$, then $z_1+z_2$ is equal to

• A. $0$
• B. purely imaginary
• C. purely real
• D. None of these

Answer 1

The correct option is A $0$

We have,

$\begin{array}{c}\left|z_1\right|=\left|z_2\right|\to \left(i\right)\\ \left(\frac{z_1}{z_2}\right)=\pi \end{array}$

from equation (i)

Squaring on both side

$\begin{array}{c}\Rightarrow {\left|z_1\right|}^2={\left|z_2\right|}^2\\ \Rightarrow {\left|z_1\right|}^2-{\left|z_2\right|}^2=0\\ \Rightarrow \left(z_1-z_2\right)\left(z_1+z_2\right)=0\\ \therefore z_1+z_2=0\end{array}$

Hence, this is the answer.