Question

If $z_1,z_2,\epsilon C$ are such that $|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$ then $\frac{z_1}{z_2}$ is

• A. zero
• B. purely real
• C. purely imaginary
• D. complex

Answer 1

The correct option is C purely imaginary

$\because |z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$

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

$\Rightarrow R\left(z_1{\overline{z}}_2\right)=0\Rightarrow z_1{\overline{z}}_2$ is imaginary

Now $\frac{z_1}{z_2}=\frac{z_1{\overline{z}}_2}{z_2{\overline{z}}_2}=\frac{\text{imaginary}}{|z_2{|}^2}=$ imaginary