Let z₁ and z₂ be two complex numbers with positive real parts satisfying z₁^2 + z₂^2 = 3 z₁ z₂ . If (z₁ z₂) = 2 and Im ( z₁ z₂ ) > 0 , then the principal argument of z₁^3 - z₂^3 is

Step-by-step solution

Given z₁^2 + z₂^2 = 3 z₁ z₂ Dividing by z₂^2 , we get ( z₁ z₂ )^2 - 3 ( z₁ z₂ ) + 1 = 0 Solving for z₁ z₂ , we get z₁ z₂ = 3 3 - 4 2 = 3 2 i 1 2 Since Im ( z₁ z₂ ) > 0 , we have z₁ z₂ = 3 2 + i 1 2 = e^ i /6 This implies | z₁ z₂ | = 1 |z₁| = |z₂| = r And ( z₁ z₂ ) = 6 (z₁) - (z₂) = 6 We are given (z₁ z₂) = 2 (z₁) + (z₂) = 2 Solving these two equations gives (z₁) = 3 and (z₂) = 6 Thus, z₁ = r e^ i /3 and z₂ = r e^ i /6 Now, z₁^3 - z₂^3 = r^3 e^ i - r^3 e^ i /2 = r^3(-1) - r^3(i) = r^3(-1 - i) The principal argument