Question

Consider the complex number $z_1$ and $z_2$ satisfying the relation $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$, then complex number $z_1{\overline{z}}_2$ is,

• A. purely real
• B. purely imaginary
• C. zero
• D. none of these

Answer 1

The correct option is B purely imaginary

${\left|z_1\right|}^2+{\left|z_2\right|}^2={\left|z_1{+z}_2\right|}^2$

${\left|z_1{+z}_2\right|}^2=\left(z_1{+z}_2\right)(\overline{z_1}+\overline{z_2})$

${⟹\left|z_1\right|}^2+{\left|z_2\right|}^2=\left(z_1{+z}_2\right)(\overline{z_1}+\overline{z_2})$

${⟹\left|z_1\right|}^2+{\left|z_2\right|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2+z_1\overline{z_2}+z_2\overline{z_1}$

$⟹z_1\overline{z_2}+z_2\overline{z_1}=0$

$⟹z_1\overline{z_2}+\overline{z_1}{z}_2=0$

$⟹Re\left(z_1\overline{z_2}\right)=0$ ..{$\because z+\overline{z}=2Re\left(z\right)$ }

$\therefore z_1\overline{z_2}$ is purely imaginary.

Ans: B