Suppose C is a NDCC corresponding
to the quadratic polynomial
$P(x,y)=Ax^2 + Bxy+Cy^2+Dx+Ey+F$ .
Definition: Let $\Delta = B^2 - 4AC$.
$\Delta$ is called the discriminant of $P$.
The Discriminant Theorem:
Proof: The proof is broken
into three steps.
- C is
an ellipse if and only if
$\Delta $ < $0$.
- C is a parabola
if and only if $\Delta = 0$.
- C is an hyperbola
if and only if $\Delta $ > $0$.
Since 1,2,and 3 are mutually exclusive and exhaustive
cases for a NDCC,
the converse statements are also true.