• The discriminant of a non-degenerate conic curve (NDCC) in the real Cartesian plane determined by a non-degenerate quadratic equation of the form
• $Ax^2 + Bxy +Cy^2 +Dx +Ey + F = 0$
is given by
$\Delta = B^2 -4AC$.

• I will present a simple proof of the classification of the conic (ellipse, parabola, or hyperbola) by the discriminant.

• The proof is based on understanding homogeneous coordinates in real projective algebraic geometry and the related 3 dimensional coordinate geometry.