Question

For complex numbers z & ω, prove that, |z|²ω−|ω|²z=z−ω if and only if ...................

A. z=w
B. zw=1
C. z=-w
D. both A and B

Answer 1

To prove |z|²ω−|ω|²z = z−ω, we need to demonstrate the equality in both directions. If this equation is true for all complex numbers z and ω, then the coefficients must be zero. Therefore, |z|² - 1 = 0, which implies |z|² = 1, and |ω|² - 1 = 0, which implies |ω|² = 1. This means that both z and ω lie on the unit circle in the complex plane.

option d is the correct

Step-by-step explanation:

To prove |z|²ω−|ω|²z = z−ω, we need to demonstrate the equality in both directions.

First, assume that |z|²ω−|ω|²z = z−ω. Simplifying this expression, we have (|z|²ω - z) - (|ω|²z - ω) = 0. Factoring out z and ω, we get z(|z|² - 1) - ω(|ω|² - 1) = 0.

If this equation is true for all complex numbers z and ω, then the coefficients must be zero. Therefore, |z|² - 1 = 0, which implies |z|² = 1, and |ω|² - 1 = 0, which implies |ω|² = 1. This means that both z and ω lie on the unit circle in the complex plane.

In conclusion, the statement |z|²ω−|ω|²z = z−ω is true if and only if both z and ω lie on the unit circle in the complex plane. This can be represented by option D. both A and B.