Question

For any two complex numbers $z_1$ and $z_2$ with $\left|z_1\right|\ne \left|z_2\right|$ ${\left|\sqrt{2}{z}_1+i\sqrt{3}{\bar{z}}_2\right|}^2+{\left|\sqrt{3}{\bar{z}}_1+i\sqrt{2}{z}_2\right|}^2$ is

• A. less than $5{\left|z_1\right|}^2+{\left|z_2\right|}^2$
• B. greater than $10\left|z_1{z}_2\right|$
• C. equal to $2{\left|z_1\right|}^2+3{\left|z_2\right|}^2$
• D. zero

Answer 1

Answer: (B)

${\left|\sqrt{2}{z}_1+i\sqrt{3}{\bar{z}}_2\right|}^2+{\left|\sqrt{3}{\bar{z}}_1+i\sqrt{2}{z}_2\right|}^2$
$=\left(\sqrt{2}{z}_1+i\sqrt{3}{\bar{z}}_2\right)\left(\sqrt{2}{\bar{z}}_1-i\sqrt{3}{z}_2\right)+\left(\sqrt{3}{\bar{z}}_1+i\sqrt{2}{z}_2\right)\left(\sqrt{3}{z}_1-i\sqrt{2}{\bar{z}}_2\right)$
$=5\left({\left|z_1\right|}^2+{\left|z_2\right|}^2\right)>5\cdot 2\sqrt{{\left|z_1\right|}^2{\left|z_2\right|}^2}$
$=10\left|z_1{z}_2\right|$
Since $A\cdot M>G\cdot M$ for $\left|z_1\right|\ne \left|z_2\right|$