How the Conic Sections Calculator works
Every conic section — circle, ellipse, parabola, or hyperbola — can be written as a single general second-degree equation in x and y. This calculator reads the six coefficients of that equation and applies the standard discriminant test to identify the curve, then computes its center and the angle needed to remove any tilt.
The general conic equation
The general form is Ax² + Bxy + Cy² + Dx + Ey + F = 0, where A, B, and C are not all zero. A, B, and C control the shape and orientation of the curve; D and E shift it left/right and up/down; F is a constant. The Bxy cross term only shows up when the curve is rotated relative to the x and y axes — for an axis-aligned conic, B is 0.
The discriminant test
The sign of the discriminant Δ = B² − 4AC classifies the conic family:
- Δ < 0: an ellipse — or a circle in the special case where A = C and B = 0.
- Δ = 0: a parabola.
- Δ > 0: a hyperbola.
This is the same test used to classify second-degree curves in analytic geometry: it looks only at the quadratic terms (A, B, C), since those terms determine the fundamental shape of the curve regardless of where it sits in the plane.
Center and rotation angle
For any conic with a center — circles, ellipses, and hyperbolas — the center (h, k) solves the linear system 2Ah + Bk = −D and Bh + 2Ck = −E, giving h = (BE − 2CD) / (4AC − B²) and k = (BD − 2AE) / (4AC − B²). Parabolas have no such center because 4AC − B² equals 0 for them. When B is not zero, the curve is rotated; the angle that eliminates the xy-term is θ = 0.5 × atan2(B, A − C), derived from the standard axis-rotation identity cot(2θ) = (A − C)/B.
A limitation to keep in mind
The discriminant test classifies the shape of the quadratic part correctly, but it does not by itself rule out degenerate cases — for example, an equation with Δ < 0 can describe a single point or have no real solutions at all, and one with Δ > 0 can describe a pair of intersecting lines instead of a true hyperbola. Confirming a genuine, non-degenerate curve requires checking the full 3×3 determinant of the equation, so treat an unexpected or oddly small result as a cue to double-check your coefficients or plot the equation.