Conic Sections Calculator - Identify Conics from General Form
Identify and classify a conic section directly from the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 using the discriminant B² − 4AC.
Enter the six coefficients A, B, C, D, E, F. The calculator reports the discriminant, the conic type (circle, ellipse, parabola, or hyperbola), and a short explanation.
Conic Sections Calculator - Identify Conics from General Form
Identify and classify a conic section directly from the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 using the discriminant B² − 4AC.
About the conic sections calculator
A conic section is the intersection of a plane with a double cone. Depending on the angle of the cut, you obtain a circle, an ellipse, a parabola, or a hyperbola. Every conic in the plane can be described algebraically by a general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, and the type of conic is determined by the sign of the discriminant Δ = B² − 4AC.
The classification rule is remarkably clean. If Δ < 0 the conic is an ellipse, with the special case A = C and B = 0 picking out a circle. If Δ = 0 the conic is a parabola. If Δ > 0 the conic is a hyperbola. There are also degenerate cases — a single point, an empty set, a single line, two parallel lines, or two intersecting lines — that arise when the equation factors in particular ways, but for non-degenerate inputs the discriminant alone is enough to identify the curve.
Why is this useful? Conics show up everywhere in science and engineering. Planetary orbits are ellipses (Kepler's first law). The path of a thrown ball, ignoring air resistance, is a parabola. The flight paths of objects escaping a gravitational field are hyperbolas. Satellite dish antennas, car headlights, and radio telescopes all exploit the reflective properties of parabolic mirrors. Whispering galleries and lithotripsy machines use the focal properties of ellipses. The cooling towers of nuclear power plants are hyperboloids. Even the design of bridges and arches relies on parabolic and catenary curves that approximate conics very closely.
The calculator is also a useful classroom tool. Students often see conics presented in standard form — (x − h)²/a² + (y − k)²/b² = 1 for an ellipse, for instance — but real problems usually present the equation already expanded into the messy general form. By entering the coefficients directly you can recover the conic type in one click, without first having to complete the square. After classification you can use the focus, directrix, and axis information from a textbook to sketch the curve or convert to standard form.
A few caveats. The discriminant test classifies only non-degenerate conics. If A = B = C = 0 the equation is linear and not a conic at all; the calculator detects this case explicitly. For exact circle detection you must have B = 0 and A = C. And when B is non-zero, the principal axes of the conic are rotated with respect to the x and y axes; the type is still determined by the discriminant, but the orientation requires diagonalising the quadratic form.
Worked examples
A few inputs that span all four conic types.
| Coefficients (A, B, C, D, E, F) | Conic type | Discriminant and notes |
|---|---|---|
| (1, 0, 1, 0, 0, −9) | Circle | Δ = 0 − 4·1·1 = −4 < 0 and A = C, B = 0. Equation x² + y² = 9 is a circle of radius 3. |
| (4, 0, 9, 0, 0, −36) | Ellipse | Δ = 0 − 4·4·9 = −144 < 0. Equation 4x² + 9y² = 36, or x²/9 + y²/4 = 1. |
| (1, 0, 0, 0, −4, 0) | Parabola | Δ = 0 − 4·1·0 = 0. Equation x² = 4y is a vertical parabola opening upward. |
| (1, 0, −1, 0, 0, −1) | Hyperbola | Δ = 0 − 4·1·(−1) = 4 > 0. Equation x² − y² = 1 is a standard rectangular hyperbola. |
| (0, 0, 0, 2, −3, 5) | Linear equation (not a conic) | All three quadratic coefficients are zero, so the equation reduces to the line 2x − 3y + 5 = 0. |
How to use the conic sections calculator
- Rearrange your equation into the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 so that the right-hand side is zero.
- Type each of the six coefficients into the matching field. Use 0 for any missing term.
- Click Identify Conic Section. The calculator reports the discriminant, the conic type, and a short explanation.
- Use the Load buttons to populate the form with canonical examples for each conic type.
- Click Reset Calculator to clear all six coefficients and start again.
Conic sections FAQ
What are the four types of conic sections?
Circles, ellipses, parabolas, and hyperbolas. They arise as the intersection of a plane with a double cone at progressively shallower angles, with the circle being the special case of a horizontal cut and the parabola the limiting case parallel to the cone's slant.
How does the discriminant classify a conic?
For the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant is Δ = B² − 4AC. If Δ < 0 the conic is an ellipse (or a circle when A = C and B = 0), if Δ = 0 it is a parabola, and if Δ > 0 it is a hyperbola.
What is a degenerate conic?
A degenerate conic is the limiting case where the equation factors into something simpler — a single point, an empty set, a single line, two parallel lines, or two intersecting lines. The discriminant test still classifies the underlying type but does not distinguish degenerate from non-degenerate.
Why is a circle a special case of an ellipse?
A circle is an ellipse with equal semi-major and semi-minor axes. In the general equation, that happens exactly when A = C and B = 0, in which case both eigenvalues of the quadratic form are equal.
What does a non-zero B coefficient mean geometrically?
A non-zero coefficient on the xy term means the conic's principal axes are rotated with respect to the coordinate axes. The conic type is still determined by the sign of B² − 4AC, but to write the equation in standard form you must first rotate the axes to eliminate the xy term.
Can the equation represent something that is not a conic?
Yes. If A, B, and C are all zero the equation is linear, representing a line or the empty set rather than a conic. The calculator detects this case and reports it explicitly.