Let z₁, z₂ and z₃ be three complex numbers on the circle |z|=1 such that (z₁) = 0 , (z₂) = 2 and (z₃) = , where (0, 2 ) . If |z₁ z ₂ + z₂ z ₃ + z₃ z ₁ |^2 = 2 and the sum of all possible values of is S , then the value of 10S is :

Step-by-step solution

Given |z|=1 , we can write the complex numbers in polar form: z₁ = e^ i(0) = 1 z₂ = e^ i 2 = i z₃ = e^ i = + i Now, compute the pairwise products: z₁ z ₂ = (1)(-i) = -i z₂ z ₃ = (i)( - i ) = + i z₃ z ₁ = ( + i )(1) = + i Summing these gives: z₁ z ₂ + z₂ z ₃ + z₃ z ₁ = ( + ) + i( + - 1) Let t = + . The expression becomes t + i(t-1) . The squared modulus is: |t + i(t-1)|^2 = t^2 + (t-1)^2 = 2t^2 - 2t + 1 We are given that this squared modulus is 2 : 2t^2 - 2t + 1 = 2 2t^2 - 2t - 1 = 0 Solving for t : t = 2 4 - 4(2)(-