Best Answer:
A general second order equation in two variables is of the form

Ax² + 2Bxy + Cy² + Dx + Ey + F = 0

for real constants A, B, C, D, E, and F. If you evaluate the discriminant B² - AC, you can determine if the equation is parabolic, hyperbolic, or elliptic. The cases are

B² - AC > 0, hyperbolic

B² - AC = 0, parabolic

B² - AC < 0, elliptic

If an equation is hyperbolic, its graph will likely be a hyperbola, parabolic a parabola, etc.

There are special (perhaps they can be considered degenerate) cases. For example, they hyperbolic equation

x² - y² = 0

has a graph that is a pair of lines (this isn't unique to this equation, I'm just using it as an example.) The elliptic equation

x² + y² = 0

has a graph that is a single point.

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