Conic Section Calculator — Identify Ax² + Bxy + Cy² + Dx + Ey + F = 0
Identify the conic Ax² + Bxy + Cy² + Dx + Ey + F = 0 — circle, ellipse, parabola, hyperbola or degenerate — with the discriminant, center or vertex, axes, eccentricity, rotation and standard form.
How it works
Discriminant Δ = B² − 4AC: Δ < 0 ellipse (circle if A = C and B = 0) · Δ = 0 parabola · Δ > 0 hyperbola · degenerate when det[[A, B/2, D/2], [B/2, C, E/2], [D/2, E/2, F]] = 0 · center from 2Ah + Bk + D = 0, Bh + 2Ck + E = 0 · rotation θ = ½·atan2(B, A − C)
Every second-degree equation in x and y describes a conic section, and the discriminant B² − 4AC identifies which one, just as it does for quadratics. The 3 × 3 determinant detects degenerate cases (a point, a pair of lines, or no real points). For an ellipse or hyperbola the calculator finds the center, moves the origin there and uses the eigenvalues of [[A, B/2], [B/2, C]] to get the semi-axes even when the conic is rotated; for axis-aligned conics (B = 0) it also completes the square to give the standard form (OpenStax Precalculus §10.4).
Worked example
4x² + 9y² − 16x + 18y − 11 = 0: B² − 4AC = 0 − 4·4·9 = −144 < 0 and A ≠ C, so it is an Ellipse. Completing the square: 4(x − 2)² − 16 + 9(y + 1)² − 9 = 11, so 4(x − 2)² + 9(y + 1)² = 36, i.e. (x − 2)²/9 + (y + 1)²/4 = 1. Center (2, −1), a = 3, b = 2, eccentricity √5/3 ≈ 0.745356.
Sources: OpenStax Precalculus 2e — 10.4 Rotation of Axes · Wolfram MathWorld — Conic Section
Frequently asked questions
How do you identify a conic section from its equation?
Compute B² − 4AC. Negative means an ellipse (a circle if A = C and B = 0), zero means a parabola and positive means a hyperbola — unless the conic is degenerate.
What is a degenerate conic?
An equation of conic form whose graph is a single point, one or two lines, or nothing at all, such as x² + y² = 0 or x² − y² = 0. The 3 × 3 determinant is 0 for these.
What does the xy term do?
A non-zero B rotates the conic. The axes are rotated by θ with tan 2θ = B/(A − C); rotating the coordinates by θ removes the xy term.
How do I put a conic in standard form?
With B = 0, group the x terms and the y terms, complete the square in each, move the constant to the right and divide so the right side is 1 (ellipse or hyperbola). For a parabola, isolate the squared variable.
Is a circle a special ellipse?
Yes — an ellipse with equal semi-axes and eccentricity 0. In the general equation that means A = C and B = 0.
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