Let z₁ and z₂ be two complex numbers such that |z₁| = 2 and |z₂| = 3 . If ( z₁ z ₂ ) = 3 and (z₁ z ₂) = 6 , then the exact value of |z₁ + z ₂|^2 + |z₁ - z₂|^2 is

Step-by-step solution

Given ( z₁ z ₂ ) = 3 (z₁) - ( z ₂) = 3 (z₁) + (z₂) = 3 Also given (z₁ z ₂) = 6 (z₁) + ( z ₂) = 6 (z₁) - (z₂) = 6 Now, |z₁ + z ₂|^2 = |z₁|^2 + | z ₂|^2 + 2|z₁|| z ₂| ( (z₁) - ( z ₂)) Since | z ₂| = |z₂| = 3 and (z₁) - ( z ₂) = (z₁) + (z₂) = 3 |z₁ + z ₂|^2 = 2^2 + 3^2 + 2(2)(3) ( 3 ) = 13 + 12 ( 1 2 ) = 19 Similarly, |z₁ - z₂|^2 = |z₁|^2 + |z₂|^2 - 2|z₁||z₂| ( (z₁) - (z₂)) |z₁ - z₂|^2 = 2^2 + 3^2 - 2(2)(3) ( 6 ) = 13 - 12 ( 3 2 ) = 13 - 6 3 Therefore, |z₁ + z ₂|^2 + |z₁ - z₂|^2 = 19 + 13 - 6 3 = 32 - 6 3 Answer: 32 -