Let z₁, z₂ and z₃ be three distinct complex numbers lying on the circle |z|=1 such that they form a geometric progression with common ratio w . If |z₁ z ₂ + z₂ z ₃ + z₃ z ₁ |^2 = 7 , then the number of possible values of the common ratio w is :

Step-by-step solution

Since z₁, z₂, z₃ are in a geometric progression with common ratio w , we have: z₂ = z₁ w z₃ = z₁ w^2 Because z₁, z₂, z₃ all lie on the unit circle |z|=1 , their moduli are 1 . |z₂| = |z₁||w| 1 = 1 |w| |w| = 1 Now, evaluate the terms in the given expression: z₁ z ₂ = z₁ (z₁ w) = z₁ z ₁ w = |z₁|^2 w = w z₂ z ₃ = (z₁ w) (z₁ w^2) = |z₁|^2 w w ^2 = w w w = |w|^2 w = w z₃ z ₁ = (z₁ w^2) z ₁ = |z₁|^2 w^2 = w^2 The expression inside the modulus simplifies to: z₁ z ₂ + z₂ z ₃ + z₃ z ₁ = w + w + w^2 = 2 w + w^2 We are given