Let z₁, z₂ C be the distinct solutions of the equation z^2 + 4z - (1 + 12i) = 0 . Then |z₁|^2 + |z₂|^2 is equal to :

Step-by-step solution

Given equation is z^2 + 4z - (1 + 12i) = 0 Let the roots be z₁ and z₂ . Sum of roots: z₁ + z₂ = -4 Product of roots: z₁ z₂ = -(1 + 12i) We know that (z₁ - z₂)^2 = (z₁ + z₂)^2 - 4z₁ z₂ (z₁ - z₂)^2 = (-4)^2 - 4(-(1 + 12i)) = 16 + 4 + 48i = 20 + 48i Taking modulus on both sides: |z₁ - z₂|^2 = |20 + 48i| = 20^2 + 48^2 = 400 + 2304 = 2704 = 52 Using the parallelogram law for complex numbers: |z₁ + z₂|^2 + |z₁ - z₂|^2 = 2(|z₁|^2 + |z₂|^2) |-4|^2 + 52 = 2(|z₁|^2 + |z₂|^2) 16 + 52 = 2(|z₁|^2 + |z₂|^2) 68 = 2(|z₁|^2 + |z₂|^