Let > 0 be a real number. Let S be the set of all complex numbers z satisfying the equation z^2 + |z|^2 - 2z = i z . If the points representing the non-zero elements of S along with the origin form a triangle in the complex plane whose area is 8 , then the value of is

Step-by-step solution

Let z = x + iy . Then z^2 = x^2 - y^2 + 2ixy and |z|^2 = x^2 + y^2 . Substitute these into the given equation: (x^2 - y^2 + 2ixy) + (x^2 + y^2) - 2(x + iy) = i (x - iy) 2x^2 - 2x + i(2xy - 2y) = y + i x Equating the real and imaginary parts: 2x(x-1) = y ... (1) 2y(x-1) = x ... (2) If x = 1 , equation (1) gives y = 0 y = 0 (since > 0 ). But substituting x=1, y=0 into (2) gives 0 = , a contradiction. Thus, x 1 . Dividing (1) by (2), we get x y = y x y^2 = x^2 y = x . Case 1: y = x Substitute into (1): 2x(x-1) = x . S