# How do use the discriminant test to determine whether the graph x^2+2xy-3y^2+5x+6y-100=0 whether the graph is parabola, ellipse, or hyperbola?

Feb 1, 2018

Given: $A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$
The discriminant is, $\partial = {B}^{2} - 4 A C$
If $\partial < 0$, then, if $B = 0 \mathmr{and} A = D$, a circle. Otherwise, an ellipse.
If $\partial = 0$, then a parabola
If $\partial > 0$, then a hyperbola.

#### Explanation:

Given: ${x}^{2} + 2 x y - 3 {y}^{2} + 5 x + 6 y - 100 = 0$

$A = 1$, $B = 2$, and $C = - 3$

$\partial = {2}^{2} - 4 \left(1\right) \left(- 3\right)$

$\partial = 16$, then it is a hyperbola.