Finding Equation of Parabola from given Focus and Directrix

1. As given in Derivation and Properties of Implicit Coordinate Equation for Axis Aligned and Arbitrarily Rotated Parabolas, for a Parabola having it's Focus at a point \((x_f,y_f)\) and having Directrix given by the equation \(A_Dx + B_Dy + C_D=0\), the Equation of Parabola is given as
   
   \({B_D}^2x^2 - 2A_DB_Dxy + {A_D}^2y^2 - 2(Px_f + A_DC_D)x - 2(Py_f + B_DC_D)y + P{x_f}^2 + P{y_f}^2 - {C_D}^2 =0\)   ...(1)
   
   where \(\mathbf{P}={A_D}^2 + {B_D}^2\)