Question

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

The correct option is B greater than $10\left|z_1{z}_2\right|$

${|\sqrt{2}{z}_1+i\sqrt{3}\overline{z_2}|}^2+{|\sqrt{3}\overline{z_1}+i\sqrt{2}{z}_2|}^2$

$=(\sqrt{2}{z}_1+i\sqrt{3}\overline{z_2})(\sqrt{2}{z}_1-i\sqrt{3}\overline{z_2})+(\sqrt{3}\overline{z_1}+i\sqrt{2}{z}_2)(\sqrt{3}\overline{z_1}-i\sqrt{2}{z}_2)$

$=5({\left|z_1\right|}^2+{\left|z_2\right|}^2)>5.2\sqrt{{\left|z_1\right|}^2.{\left|z_2\right|}^2}=10\left|z_1{z}_2\right|$

Since $A.M.>G.M>$ for $\left|z_1\right|\ne \left|z_2\right|$