Question

If z and w be two complex numbers, such that
$|z{|}^{2}w-|w{|}^{2}z=z-w$ and $(z\ne w),$ then

• A. $z=\overline{w}$
• B. $z\overline{w}=1$
• C. $\overline{z}w=2$
• D. $z\overline{w}=4$

Answer 1

The correct option is B $z\overline{w}=1$

$|z{|}^{2}w+w=z+|w{|}^{2}z$
$\frac{w}{z}=\frac{|w{|}^2+1}{|z{|}^2+1},\frac{\overline{w}}{\overline{z}}=\frac{|w{|}^2+1}{|z{|}^2+1}$
$\therefore \frac{\overline{w}z}{\overline{z}w}=1$
$\Rightarrow \overline{w}z=1=\overline{z}w$