Question

If arg $\left(Z_1+Z_2\right)=0$ and $|z_1|=|z_2|=1.$ then.

• A. $z_1+z_2=0$
• B. $z_1{\overline{z}}_2=1$
• C. $z_1={\overline{z}}_2$
• D. none

Answer 1

Answer: (C)

$z_1={\overline{z}}_2$

We have,

$\arg \left(Z_1+Z_2\right)=0\left|Z_1\right|=1\left|Z_2\right|=1$

$\begin{array}{c}{Z}_1+Z_2=k\left(k>0\right)\\ Z_1=\cos \theta +i\sin \theta \\ Z_2=\cos \theta +i\sin \theta \end{array}$

$\cos \theta +\cos \theta +\left(i\sin \theta +i\sin \theta \right)=k$

$\Rightarrow \sin \phi =-\sin \theta$

$\Rightarrow \cos \phi =-\cos \theta as\left|Z_2\right|=1$

$Z_1=\cos \theta +i\sin \theta$

$Z_2=\cos \theta -i\sin \theta$

$Z_1=\overline{Z_2}$

Then,

Option $C$ is correct answer.