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a y^2 + b xy + c x^2 + d y + e x + f = 0

can be 1) a parabola, an ellipse, an hyperbola or 2) an empty set, a line, two intersecting lines, two parallel lines, a circle, a point, and pherhaps something else...

I want two know which of these will I get.

I know the rule b^2-4ac. but the degenerate cases deceive me too often. Is there a method to decide which set the quadratic equation describes? Of course you can try completing squares in several ways, but that takes lots of trials and thought, doesn't it? is there a better way?

thanks!