General Equation of a Conic

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