Question

Let z and ω be complex numbers such that ω has non-zero imaginary part and z≠1. If the expression (ω-\(\overline{\omega}\)z)/(1-z) is purely real, then find the set of values of z.

(a) {z : |z|=1}

(b) {z : z=z̅}

(c) {z : z≠1}

(d) {z : |z|=1, z≠1}

I had been asked this question during an online exam.

This intriguing question originated from Complex Conjugates topic in section Complex Numbers of Engineering Mathematics

Answer 1

Right choice is (d) {z : |z|=1, z≠1}

The best I can explain: (ω-\(\overline{\omega}\)z)/(1-z) = (\(\overline{\omega}\)-ωz̅)/(1-z̅)⇒(zz̅-1)(\(\overline{\omega}\)-ω) = 0

⇒zz̅=1⇒|z|^2=1⇒|z|=1.