Question

If $Z_1,Z_2$ are complex numbers such that $Z_1+Z_2,Z_1{Z}_2$ are real numbers then?

• A. $z_2={\overline{z}}_1$
• B. $z_1=z_2$
• C. $z_1+z_2=0$
• D. $z_1-2z_2=0$

Answer 1

The correct option is A $z_2={\overline{z}}_1$

$for{z}_1{z}_2$ to the real
$z_2=\overline{z_1}or\overline{-z_1}$
$\Rightarrow z_1{z}_2=z_1{z}_1={\left|z_1\right|}^2$
$or{z}_1{z}_2=-{\left|z_1\right|}^2$
also
$z_1+z_2=real$
which isn't possible for $z_2=-\overline{z_1}$
$for{z}_1-\overline{z_1}=im\left(z_1\right)$
$So,z_2=\overline{z_1}$