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.