How do use the discriminant test to determine whether the graph $4 x^{2} - 8 x y + 6 y^{2} + 10 x - 2 y - 20 = 0$ $4x^2-8xy+6y^2+10x-2y-20=0$ 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

How do use the discriminant test to determine whether the graph 4x^2-8xy+6y^2+10x-2y-20=04x2−8xy+6y2+10x−2y−20=04x^2-8xy+6y^2+10x-2y-20=0 whether the graph is parabola, ellipse, or hyperbola?