Question

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

• A. $1$
• B. $3$
• C. $0$
• D. none of these

Answer 1

Answer: (C)

$0$

$Let{z}_1=r{e}^{i{\theta }_1},z_2=r{e}^{i{\theta }_2}arg\left(\frac{z_1}{z_2}\right)=arg\left(\frac{r{e}^{i{\theta }_1}}{r{e}^{i{\theta }_2}}\right)=arg\left(r{e}^{i({\theta }_1-{\theta }_2)}\right)={\theta }_1-{\theta }_2=\pi \left(Given\right)\therefore {\theta }_1=\pi +{\theta }_2{z}_1=r{e}^{i{\theta }_1}=r{e}^{i(\pi +{\theta }_2)}=r{e}^{-i{\theta }_2}=-r{e}^{i{\theta }_2}\therefore z_1+z_2=-r{e}^{i{\theta }_2}+r{e}^{i{\theta }_2}=0$