Question

For any two complex numbers $z_1,z_2$ we have ${|z_1+z_2|}^2={\left|z_1\right|}^2+{\left|z_2\right|}^2$, then

• A. $Re\left(\frac{z_1}{z_2}\right)=0$
• B. $Im\left(\frac{z_1}{z_2}\right)=0$
• C. $Re\left(z_1{z}_2\right)=0$
• D. $Im\left(z_1{z}_2\right)=0$

Answer 1

Answer: (A)

$Re\left(\frac{z_1}{z_2}\right)=0$

$|z_1+z_2{|}^2=|z_1{|}^2+|z_2{|}^2$
$|z_1{|}^2+|z_2{|}^2+2Re\left(z_1{\overline{z}}_2\right)=|z_1{|}^2+|z_2{|}^2$
$Re\left(z_1{\overline{z}}_2\right)=0$
$Re\left(\frac{z_1}{z_2}|z_2{|}^2\right)=0$
$Re\left(\frac{z_1}{z_2}\right)=0$
Hence, option A.