Free Conic Sections Calculator

Last updated October 10, 2026 · Report an error

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Conic sections are the family of curves — circles, ellipses, parabolas, and hyperbolas — formed by slicing a double cone with a flat plane at different angles, which is where their name comes from. Every one of these curves can also be described algebraically by a single general second-degree equation, Ax² + Bxy + Cy² + Dx + Ey + F = 0, and remarkably, the specific curve type is determined entirely by just three of those coefficients: A, B, and C. Conic sections are foundational across science and engineering: planetary and satellite orbits are ellipses (or, for objects that escape a star's gravity, parabolas or hyperbolas), parabolic reflectors focus light or radio waves in headlights, satellite dishes, and telescopes, and architectural and civil engineering designs use elliptical and parabolic arches for their structural and acoustic properties. Being able to classify a curve directly from its equation, without graphing it first, is a standard step in analytic geometry and calculus courses.

How it works

Enter the six coefficients A through F from the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. The calculator computes the discriminant, B² − 4AC, which measures how the curve's squared terms interact: a negative discriminant means an ellipse (or specifically a circle, if additionally A = C and B = 0, since a circle is just a perfectly symmetric ellipse), a discriminant of exactly zero means a parabola, and a positive discriminant means a hyperbola. This works because the discriminant is directly tied to how a rotated coordinate system would need to be chosen to eliminate the xy cross-term and reveal the curve's standard form.

Enter a (coefficient of x²).
Enter b (coefficient of xy).
Enter c (coefficient of y²).
Enter d (coefficient of x).
Enter e (coefficient of y).
Enter f (constant).
Click Calculate to see your results.

Examples

x² + y² − 9 = 0

A=1, B=0, C=1. Discriminant = 0² − 4(1)(1) = -4 < 0, and since A = C and B = 0, this is specifically a circle — here, one of radius 3 centered at the origin.

4x² + 9y² − 36 = 0

A=4, B=0, C=9. Discriminant = 0² − 4(4)(9) = -144 < 0, so this is an ellipse — but since A ≠ C, it's not a circle, matching the standard ellipse form x²/9 + y²/4 = 1.

y² − 4x = 0

A=0, B=0, C=1. Discriminant = 0² − 4(0)(1) = 0, so this is a parabola — matching the familiar form y² = 4x, which opens rightward along the x-axis.

Common mistakes to avoid

Forgetting to include the xy cross-term coefficient B, which is essential for correctly detecting rotated conics.
Assuming any equation with a negative discriminant is automatically a circle — it's only a circle if the additional conditions A = C and B = 0 also hold.
Treating the discriminant test as a complete classification, missing that degenerate cases (single points, one or two lines) can occur at the same boundary values.
Mixing up the sign convention — a common error is computing 4AC − B² instead of B² − 4AC, which flips the ellipse/hyperbola classification.

Frequently asked questions

No — the discriminant test classifies the general curve type based on the squared-term coefficients alone, but it doesn't separately check whether the specific D, E, F values happen to produce a degenerate case, such as a single point, one line, or two intersecting lines, which can occur at the same discriminant boundaries.

It classifies the curve as a parabola (or, in a degenerate case, a single line or two parallel lines) — the boundary case between the "closed" ellipse family (negative discriminant) and the "open," two-branched hyperbola family (positive discriminant).

A nonzero B term indicates the conic is rotated relative to the x- and y-axes. Without factoring B into the discriminant B² − 4AC, a rotated ellipse could be misclassified, since looking at A and C alone doesn't account for that rotation.

A circle is simply a special case of an ellipse where the two axes are equal in length — algebraically, that corresponds to A = C and B = 0 in the general equation, in addition to the discriminant already being negative.

Yes — depending on D, E, and F, some equations with a negative discriminant describe an "imaginary" ellipse with no real (x, y) solutions at all, such as x² + y² + 1 = 0, which the discriminant test alone won't catch.

Under an inverse-square gravitational force, the trajectory of any orbiting or passing body is mathematically forced into one of the four conic shapes — a closed elliptical (or circular) orbit for bound objects, or an open parabolic/hyperbolic path for an object with exactly escape velocity or more.