Conic section analyzer
Classify a second-degree equation in x and y as a circle, ellipse, parabola, hyperbola or degenerate case. Exact invariants, the center, axes, foci and vertices are reported.
Equations are limited to 1,000 characters and 100 digits per number, with exact values up to 1,024 digits. Only x, y and numbers are allowed. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report and a CSV of key points. Use x and y with powers up to 2, or six numbers A, B, C, D, E, F. Up to 1,000 characters. Review the settings before running.
Your result
How to use Conic section analyzer
- Type the equation, such as 9x^2 + 4y^2 - 36x + 8y + 4 = 0, or six coefficients A, B, C, D, E, F.
- Run the analysis.
- Read the classification, invariants, center, axes and key points.
- Download the JSON report or the CSV of points.
Useful next steps
How the curve is classified
The equation is rewritten as Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0. The discriminant B^2 - 4AC and the determinants of the 2 by 2 and 3 by 3 coefficient matrices decide the type, including degenerate cases such as a point, two lines or no real points. Method: OpenStax, Precalculus 2e, section 10.4 Rotation of Axes (2021).
Exact and decimal values
Invariants, the center and the eigenvalues are exact fractions or square roots. When B is 0 the axes, foci, vertices, asymptotes and directrix are exact too, and the standard form puts a fractional denominator in parentheses, as in y^2/(1/4) = 1, so it reads correctly from left to right. A parabola's vertex, focus and focal length (the distance from vertex to focus, always positive) are exact even when the curve is rotated, as long as each exact value, and every step that produces it, fits in 1,024 digits. Otherwise, they are computed exactly and then rounded to 10 significant digits, marked as not exact. For a rotated ellipse or hyperbola, foci and vertices are decimals with 10 significant digits and are marked as not exact.