Question

If $z_1$ and $z_2$ are any two complex numbers then ${\left|z_1+z_2\right|}^2+{\left|z_1-z_2\right|}^2$ is equal to

• A. $2{\left|z_1\right|}^2{\left|z_2\right|}^2$
• B. $2{\left|z_1\right|}^2+2{\left|z_2\right|}^2$
• C. ${\left|z_1\right|}^2+{\left|z_2\right|}^2$
• D. $2\left|z_1\right|\left|z_2\right|$

Answer 1

Answer: (B)

${\left|z_1+z_2\right|}^2+{\left|z_1-z_2\right|}^2$
$={\left|z_1\right|}^2+{\left|z_2\right|}^2+z_1{\bar{z}}_2+{\bar{z}}_1{z}_2+{\left|z_1\right|}^2+{\left|z_2\right|}^2-z_1{\bar{z}}_2-{\bar{z}}_1{z}_2$
$=z\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)$