Let F(x1, y1) be the focus, lx + my + n = 0, the equation of the directrix and e the eccentricity of the conic. Let P(x, y) be any point on the conic.
FP/PM = e
FP2 = e2 × PM2
Simplifying this, you get an equation of second degree in x and y of the form
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0