The number of complex numbers z satisfying the equation |z - 2| = z - 4 + 3i is

Step-by-step solution

Let z = x + iy , where x, y R . Substituting this into the given equation, we get: |x - 2 + iy| = x + iy - 4 + 3i (x-2)^2 + y^2 = (x - 4) + i(y + 3) Since the left hand side is purely real, the imaginary part of the right hand side must be zero. y + 3 = 0 y = -3 Equating the real parts, we get: (x-2)^2 + (-3)^2 = x - 4 x^2 - 4x + 13 = x - 4 Squaring both sides: x^2 - 4x + 13 = (x - 4)^2 x^2 - 4x + 13 = x^2 - 8x + 16 4x = 3 x = 3 4 However, we must check if this solution satisfies the original radical equation. The