Discriminant
Just as non-degenerate real conics can be classified by the discriminant of their imaginary part, considered as a quadratic form in (the determinant of the matrix of the associated symmetric form), a conic is degenerate if and only if the discriminant of the homogeneous quadratic form in is zero, where the affine equation
(factors of 2 for cross terms) is homogenized to
the discriminant in this sense is then the determinant of the matrix:
Recall that the discriminant for the elliptic/parabolic/hyperbolic is the determinant of the matrix:
Read more about this topic: Degenerate Conic