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Given an equation, we can determine which type of conic section it describes by calculating ##d=B^2-4AC## (see attachment). However, the theorem demands that the equation describes a conic section. So how do we show that it does?

Does there exist a counterexample where ##B^2-4AC<0##, but it is neither a circle or an ellipse?

EDIT: I found that if ##d<0## there is no need to check that a conic section exists because ##d<0## implies a conic section exists.

But it remains to find out whether there is a need to check if a conic section exists when ##d=0## or ##d>0##, given that ##B\neq0##.

Does there exist a counterexample where ##B^2-4AC<0##, but it is neither a circle or an ellipse?

EDIT: I found that if ##d<0## there is no need to check that a conic section exists because ##d<0## implies a conic section exists.

But it remains to find out whether there is a need to check if a conic section exists when ##d=0## or ##d>0##, given that ##B\neq0##.

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