conic is a parabola and its eccentricity equals 1 (provided it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or ellipse, the eccentricity is given by

where if the determinant of the 3 × 3 matrix above is negative and if that determinant is positive.

It can also be shown that the eccentricity is a positive solution of the equation

where again This has precisely one positive solution--the eccentricity-- in the case of a parabola or ellipse, while

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