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
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.