Complex Numbers

For any complex number w = c + id, let \(\arg (w) \in ( - \pi ,\pi ]\), where \(i = \sqrt { - 1}\). Let \(\alpha\) and \(\beta\) be real numbers such that for all complex numbers z = x + iy satisfying \(\arg \left( {{{z + \alpha } \over {z + \beta }}} \right) = {\pi \over 4}\), the ordered pair (x, y) lies on the circle \({x^2} + {y^2} + 5x - 3y + 4 = 0\), Then which of the following statements is (are) TRUE?