If z 1 , z 2 are complex numbers such that Re z 1 = z 1 - 1 and Re z 2 = z 2 - 1 and arg z 1 - z 2 = π 6 , then I m z 1 + z 2 is equal to :

Step-by-step solution

Let z 1   = x 1 + i y 1 and z 2 = x 2 + i y 2 Given z 1 - 1 = R e z 1 ⇒ x 1 - 1 2 + y 1 2 = x 1 2 ⇒ y 1 2 - 2 x 1 + 1 = 0   . . . . . . . . . . . . . . 1 z 2 - 1 = R e z 2 ⇒ x 2 - 1 2 + y 2 2 = x 2 2 ⇒ y 2 2   - 2 x 2 + 1 = 0   . . . . . . . . . . . 2 Subtracting equation 2 from equation 1 , we get y 1 2 - y 2 2 - 2 x 1 - x 2 = 0 ⇒ y 1 - y 2   y 1 + y 2 = 2 x 1 - x 2 ⇒ y 1 + y 2 = 2 x 1 - x 2 y 1 - y 2   . . . . . . . . . . . . 3 Also given, a r g &#