Conic Section Calculator

Enter the five coefficients of A x^2 + C y^2 + D x + E y + F = 0 and this completes the square to say which conic section the equation describes. It returns the standard form, the centre or vertex, the vertices, the foci, the eccentricity and the axis lengths, plus the asymptotes of a hyperbola or the focus and directrix of a parabola. Degenerate cases are named rather than dressed up as curves: a single point, an empty graph, a pair of crossing lines or a pair of parallel lines. An xy term is not accepted, because a rotated conic needs its rotation angle reported alongside the centre and that is not worked out here.