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?

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Answer 1 · 2016-12-31T12:46:32

See the classification, in the explanation.

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

second degree

#ax^2+2hxy+by^2+2gx+2fy+c=0#

represents a real circle, if

#a=b, h== and g^2+f^2>0#

The graph is a pair of straight lines, if

#abc+2fgh-af^2-bg^2-ch^2=0#.

Failing these tests, the graph is

an ellipse, if

#D = h^2-ab < 0#,

a parabola, if

D = 0

and a hyperbola, if

#D>0#.

Here, the equation is

#4x^2-8xy+6y^2+10x-2y-20=0#.

#D=(-4)^2-(4)(6)= -8<0#

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

See the Socratic graph for the ellipse