Let z and w be two complex numbers such that w = z z ¯ - 2 z + 2 , z + i z - 3 i = 1 and Re w has minimum value. Then, the minimum value of n ∈ N for which w n is real, is equal to _______.

Step-by-step solution

ω = z z ¯ - 2 z + 2 z + i z - 3 i = 1 ⇒ z + i = z - 3 i ⇒ x 2 + y + 1 2 = x 2 + y - 3 2 ⇒ y = 1 ⇒ z = x + i ,   x ∈ R ω = x + i x - i - 2 x + i + 2 = x 2 + 1 - 2 x - 2 i + 2 Re ω = x 2 - 2 x + 3 = x - 1 2 + 2 For min Re ω ,   x = 1 ⇒ ω = 2 - 2 i = 2 1 - i = 2 2 e - i π 4 ω n = 2 2 n e - i n π 4 For real and minimum value of n , n = 4