Let w I m w ≠0 be a complex number. Then, the set of all complex numbers z satisfying the equation w - w ¯ z = k 1 - z , for some real number k , is

Step-by-step solution

w ∈ C , I m   w ≠ 0  ⇒ w = a + i b : b ≠0 (suppose) w - w ¯   z = k 1 - z If z = x + i y ⇒ a + i b - a - i b x + i y = k 1 - x + i y ⇒ a + i b - a x + i a y - i b x + b y = k - k x - i k y Equating real and imaginary parts a - a x - b y = k - k x         … ( 1 ) and b - a y + b x = - k y           … 2 From 1   k - a x - 1 = b y 2   k - a y = b ( x + 1 ) Dividing the two equations, we get, x - 1