# How do use the discriminant test to determine whether the graph 4xy+5x-10y+1=0 whether the graph is parabola, ellipse, or hyperbola?

Dec 22, 2016

#### Explanation:

Here is a reference Conic Sections that I will use.

Here is the general Cartesian form of a conic section:

$A {x}^{2} + B x y + C {y}^{2} + D x + E y + F = 0$

The discriminant is: ${B}^{2} - 4 A C$

The discriminant test is the following 3 "If-then" tests; two of which have subordinate special cases:

$\text{ }$If ${B}^{2} - 4 A C < 0$, then the equation represents an ellipse.

$\text{[1.1] }$A subordinate special case of this occurs when $A = C \mathmr{and} B = 0$, then the equation represents a circle.

$\text{ }$If ${B}^{2} - 4 A C = 0$, then the equation represents a parabola.

$\text{ }$If ${B}^{2} - 4 A C > 0$, then the equation represents a hyperbola.

$\text{[3.1] }$A subordinate special case of this occurs, when $A + C = 0$, then the equation represents a rectangular hyperbola.

The given equation is the type specified by [3.1]. A rectangular hyperbola .

Here is the graph of the equation: 