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?

1 Answer
Dec 31, 2016

See the classification, in the explanation.

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