Question

For complex numbers z & $\omega$, prove that, $|z{|}^2\omega -|\omega {|}^{2}z=z-\omega$ if and only if ...................

• A. $z=\omega$
• B. $z\overline{\omega }=1$
• C. $z=-\omega$
• D. both A and B

Answer 1

The correct option is D both A and B

Comparing both sides we get

$|z{|}^2.w=z$ ...(i)

and

$|w{|}^2.z=w$ ...(ii)

Considering i, we get

$z.\overline{z}=\frac{z}{w}$

$z.\overline{z}=z.\overline{w}$

Hence

$\overline{z}=\overline{w}$

Or
$z=w$

$\frac{z}{w}=1$

Or
$z\overline{w}=1$