COMPLEX NUMBERS QUESTION

Question

Let z1 and z2 be two complex numbers such that (z1-2z2)/(2-z1*conjug(z2)) is unimodular. If z2 is not unimodular, then find |z1|

Answer 1

Let's denote y=conjugate z2

(z₁-2z₂)/(2-z₁y)*Conjugate[(z₁-2z₂)/(2-z₁y)]=1

If you multiply, you get

|z₁|²-2Re(z₁y)+4|z₂|²=4-2Re(z₁y)+|z₁|²|z₂|²

From here

|z₁|²(1-|z₂|²)=4(1-|z₂|²).

Since z₂ is not unimodular you can divide both sides by 1-|z₂|²≠0

|z₁|²=4.

Note: When writing you should be careful because of indexes and conjugates it is easy to make mistakes and at once you can find mistakes in my asnwer as well.