Let α = 8 - 14 i , A = z ∈ ℂ : α z - α ¯ z ¯ z 2 - ( z ¯ ) 2 - 112 i = 1 and B = z ∈ ℂ : z + 3 i = 4 Then, ∑ z ∈ A ∩ B ( Re z - I m z ) is equal to ________

Step-by-step solution

Given: A = z ∈ ℂ : α z - α ¯ z ¯ z 2 - ( z ¯ ) 2 - 112 i = 1 B = z ∈ ℂ : z + 3 i = 4 α = 8 - 14 i Let z = x + i y So, α z = 8 - 14 i x + i y ⇒ α z = ( 8 x + 14 y ) + i ( - 14 x + 8 y ) Also, z + z ¯ = 2 x and z - z ¯ = 2 i y Set A: α z - α ¯ z ¯ z 2 - ( z ¯ ) 2 - 112 i = 1 ⇒ 2 i - 14 x + 8 y z + z ¯ z - z ¯ - 112 i = 1 ⇒ 2 i - 14 x + 8 y 4 i x y - 112 i = 1 ⇒ - 14 x + 8 y 2 x y - 56 =