Classify any axis-aligned conic from its general-form coefficients. See the discriminant test, the completing-the-square work, the standard form, and the eccentricity with the centre, vertices, foci, directrix, and asymptotes.
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The Conic Section Classifier takes the five coefficients of the general form A x² + C y² + D x + E y + F = 0 and identifies the curve with the discriminant test B² − 4AC, then completes the square to reach the standard form.
Every result reports the eccentricity formula e = c/a alongside the centre or vertex, the semi-axes a and b, the focal distance c, the foci, the vertices and co-vertices, the parabola's directrix, and the hyperbola's asymptotes. Degenerate cases — a single point, intersecting lines, parallel lines, a single line, or no real graph at all — are diagnosed and explained rather than silently mis-classified.
The eccentricity formula for a central conic is e = c/a, where a is the semi-major (or semi-transverse) axis and c is the centre-to-focus distance. A circle has e = 0, an ellipse 0 < e < 1, a parabola exactly e = 1, and a hyperbola e > 1.
Compute the discriminant B² − 4AC. For an axis-aligned conic B = 0, so the test is just −4AC: negative gives the ellipse family (a circle when A = C), positive gives a hyperbola, and zero gives a parabola.
Put the ellipse in standard form, read a² and b² with a > b, then use c² = a² − b² to find c. The eccentricity is e = c/a, always strictly between 0 and 1.
For a hyperbola use c² = a² + b², where a is the semi-transverse axis and b the semi-conjugate axis. Then e = c/a, which is always greater than 1.
A degenerate conic appears when completing the square leaves a constant of zero or the wrong sign. The graph collapses to a single point, two intersecting lines, two parallel lines, one line, or no real points at all. Eccentricity is undefined for every degenerate case.
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