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

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Answer 1 · 2016-12-22T18:04:14

Please see the explanation.

Here is a reference Conic Sections that I will use.

Here is the general Cartesian form of a conic section:

$Ax^2+Bxy+Cy^2+Dx+Ey+F=0$

The discriminant is: $B^2 - 4AC$

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

$"[1] "$ If $B^2 - 4AC < 0$ , then the equation represents an ellipse.

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

$"[2] "$ If $B^2 - 4AC = 0$ , then the equation represents a parabola.

$"[3] "$ If $B^2 - 4AC > 0$ , then the equation represents a hyperbola.

$"[3.1] "$ 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:

enter image source here

How do use the discriminant test to determine whether the graph 4xy+5x-10y+1=04xy+5x-10y+1=0 whether the graph is parabola, ellipse, or hyperbola?