Through a fixed point \( P(\alpha, \beta) \) a variable line is drawn to cut the coordinate axes at \( A \) and \( B \). The locus of the mid-point of \( A B \) is
(A) a hyperbola with eccentricity 2
(B) a hyperbola with centre \( \left(\frac{\alpha}{2}, \frac{\beta}{2}\right) \)
(C) a hyperbola with asymptotes along axes
(D) not a hyperbola
