Examples
As an example of the first failure, reducibility, is not degenerate (it defines a hyperbola), but is degenerate because it is reducible – it factors as, and corresponds to two intersecting lines or an "X".
As an example of the second failure, not enough points (over the field of definition), over the real numbers is not degenerate (it defines a circle) but is degenerate – it defines a point, not a curve, and is likewise degenerate, defining the empty set. Note that over the complex numbers factors as and is degenerate because reducible, while defines a non-degenerate curve, an ellipse/hyperbola (these are not distinct over the complex numbers, because there is no sense of positive or negative).
Read more about this topic: Degenerate Conic
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