Question

Find the value of $k$, if for the complex numbers $z_1$ and $z_2$
${1-\left|\overline{z_1}{z}_2\right|}^2-{|z_1+z_2|}^2=k(1-{\left|z_1\right|}^2)(1-{\left|z_2\right|}^2)$

• A. $2$
• B. $3$
• C. $1$
• D. $4$

Answer 1

Answer: (A)

$2$

Given $1-{\left|\overline{z_1}{z}_2\right|}^2+(-{\left|\overline{z_1}{z}_2\right|}^2)=k(1-{\left|z_1\right|}^2)(1-{\left|z_2\right|}^2)$

$\Rightarrow 1-{\left|z_1\right|}^2{\left|z_2\right|}^2-({\left|z_1\right|}^2+{\left|z_2\right|}^2+2\left|z_1\right|\left|z_2\right|)=k(1-{\left|z_1\right|}^2-{\left|z_2\right|}^2+{\left|z_1\right|}^2{\left|z_2\right|}^2)$

$\Rightarrow {\left|\overline{z_1}\right|}^2={\left|z_1\right|}^2$

For $k=2$ the equation satisfy.

Hence, the answer is $2.$