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:

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