Question

If $z_1$ and $z_2$ are two complex numbers such that $Im(z_1+z_2)=0$, $Im\left(z_1{z}_2\right)=0$, then.

• A. $z_1=-z_2$
• B. $z_1=z_2$
• C. $z_1=\overline{z_2}$
• D. None of these

Answer 1

Answer: (C)

$z_1=\overline{z_2}$

Since, Im$(z_1+z_2)=0$ and Im$\left(z_1{z}_2\right)=0$ .... $\left(i\right)$
$\Rightarrow z_1+z_2$ and $z_1{z}_2$ both are real.
Let $z_1=a_1+i{b}_1,z_2=a_2+i{b}_2$
Then, $z_1+z_2$ is real $\Rightarrow b_2=-b_1$
$z_1{z}_2$ is real i.e. Im$\left(z_1{z}_2\right)=0$
$\Rightarrow a_1{b}_2+a_2{b}_1=0$
$\Rightarrow -a_1{b}_1+a_2{b}_1=0$ ...... $[\because b_2=-b_1]$
$\Rightarrow a_1=a_2$
So, $z_1=a_1+i{b}_1=a_2-i{b}_2=\overline{z_2}$.

Hence, $z_1=\overline{z_2}$