# How do use the discriminant test to determine whether the graph 4x^2-8xy+6y^2+10x-2y-20=0 whether the graph is parabola, ellipse, or hyperbola?

Dec 31, 2016

See the classification, in the explanation.

#### Explanation:

graph{4x^2-8xy+6y^2+10x-2y-20=0 [-20, 20, -10, 10]} $' D = {h}^{2} - a b '$ is called the discriminant of the general equation of the

second degree

$a {x}^{2} + 2 h x y + b {y}^{2} + 2 g x + 2 f y + c = 0$

represents a real circle, if

$a = b , h = = \mathmr{and} {g}^{2} + {f}^{2} > 0$

The graph is a pair of straight lines, if

$a b c + 2 f g h - a {f}^{2} - b {g}^{2} - c {h}^{2} = 0$.

Failing these tests, the graph is

an ellipse, if

$D = {h}^{2} - a b < 0$,

a parabola, if

D = 0

and a hyperbola, if

$D > 0$.

Here, the equation is

$4 {x}^{2} - 8 x y + 6 {y}^{2} + 10 x - 2 y - 20 = 0$.

$D = {\left(- 4\right)}^{2} - \left(4\right) \left(6\right) = - 8 < 0$

The graph ( if not a pair of straight lines ) is an ellipse.

See the Socratic graph for the ellipse