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the following formula gives the eccentricity if the conic section is not a parabola (which has eccentricity equal to 1), not a degenerate hyperbola or degenerate ellipse, and not an imaginary ellipse:

:

where if the determinant of the 3×3 matrix

:

is negative or if that determinant is positive.

Ellipses

The eccentricity of an ellipse is strictly less than 1. When circles (which

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