**Definition: Let $\Delta = B^2 - 4AC$.
$\Delta$ is called the discriminant of $P$.
The Discriminant Theorem: **

- 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 $ > $0$.

the converse statements are also true.

EOP.