Let a , b , x and y be real numbers such that a - b = 1 and y ≠ 0 . If the complex number z = x + i y satisfies I m a z + b z + 1 = y , then which of the following is (are) possible value (s) of x ?

Step-by-step solution

I m a z + b z + 1 = y and z = x + i y ∴       I m a x + i y + b x + i y + 1 = y multiply and divide by x + 1 − i y and equate imaginary part ⇒     - y a x + b + a y x + 1 = y x + 1 2 + y 2 ⇒         a - b y = y x + 1 2 + y 2 ∵         y ≠ 0   a n d   a - b = 1 ⇒         x + 1 2 + y 2 = 1 ⇒         x = - 1 ±   1 - y 2