For all complex numbers z of the form 1 + i α , α ∈ R , if z 2 = x + i y , then

Step-by-step solution

Given z = 1 + iα ,   z 2 = x + iy ⇒ 1 + iα 2 = x + iy ⇒ 1 2 + i 2 α 2 + 2 i α = x + i y Using i 2 = - 1 , ⇒ 1 - α 2 + 2 i α = x + iy Equating the real and imaginary parts, we get x = 1 - α 2 and y = 2 α ⇒ α 2 = 1 - x and y 2 = 4 α 2 ⇒ y 2 = 4 1 - x ⇒ y 2 = 4 - 4 x ⇒ y 2 + 4 x - 4 = 0 .