If focus is (alpha, beta) and directrix is lx+my+n=0, then the parabola equation is (x-alpha)^2+(y-beta)^2 = (lx+my+n)^$\frac{2}{l^2+m^2}$. This comes directly from the definition: distance from focus = distance from directrix. This form is useful when the parabola has a tilted axis (not parallel to coordinate axes).