Canonical form of a conics
Find the congruence which transform given conic to the axis form - centre coincide with origin O=(0,0), major and minor axis with coordinate axis x and y.
Determinant C of the quadratic part has the same signum for all mutually affine (and also congruent) conic section. Affine map preserves signum of determinant C, thus it preserves the signature of quadratic part.