Let a complex number z = x+iy satisfy the conditions ( z-1 z+1 ) = 4 and |z - 2| = 2|z - i| . The value of [x] + [y] , where [t] denotes the greatest integer function, is :

Step-by-step solution

Given ( z-1 z+1 ) = 4 ( (x-1+iy)(x+1-iy) (x+1)^2+y^2 ) = 4 ( x^2+y^2-1 + i(2y) (x+1)^2+y^2 ) = 4 For the argument to be 4 , the real and imaginary parts must be equal and strictly positive. x^2 + y^2 - 1 = 2y and 2y > 0 x^2 + y^2 - 2y - 1 = 0 with y > 0 ... (1) Also given |z - 2|^2 = 4|z - i|^2 (x-2)^2 + y^2 = 4(x^2 + (y-1)^2) x^2 - 4x + 4 + y^2 = 4x^2 + 4y^2 - 8y + 4 3x^2 + 3y^2 + 4x - 8y = 0 ... (2) From (1), we have x^2 + y^2 = 2y + 1 . Substituting this into (2): 3(2y + 1) + 4x - 8y = 0 6y + 3 + 4x - 8y = 0 y =