The number of complex numbers z satisfying the equations z^2 + z ^2 = |z|^2 and |z - 2| = 2 simultaneously is :

Step-by-step solution

Let z = x + iy , where x, y R . From the first equation z^2 + z ^2 = |z|^2 : (x + iy)^2 + (x - iy)^2 = x^2 + y^2 (x^2 - y^2 + 2ixy) + (x^2 - y^2 - 2ixy) = x^2 + y^2 2x^2 - 2y^2 = x^2 + y^2 x^2 = 3y^2 From the second equation |z - 2| = 2 : |x - 2 + iy|^2 = 4 (x - 2)^2 + y^2 = 4 x^2 - 4x + 4 + y^2 = 4 x^2 + y^2 = 4x Substitute y^2 = x^2 3 into the second equation: x^2 + x^2 3 = 4x 4x^2 3 = 4x x^2 - 3x = 0 x(x - 3) = 0 This gives x = 0 or x = 3 . Case 1: If x = 0 , then y^2 = 0 y = 0 . This gives the solution z = 0 .