Conic Section Identifier Calculator

Work out conic section identifier instantly with clear inputs, formula shown and shareable results.

About Conic Section Identifier Calculator

For the general second-degree curve Ax² + Bxy + Cy² + Dx + Ey + F = 0, the sign of the discriminant B² − 4AC fixes the type: negative gives an ellipse (a circle when A = C and B = 0), zero a parabola, positive a hyperbola.

Formulas

Discriminant test

B² − 4AC < 0 ellipse, = 0 parabola, > 0 hyperbola

Axis rotation

θ = ½·atan2(B, A − C)

Worked example

With the default values below, Conic Section Identifier Calculator returns Conic type of Ellipse. Change any input above and every figure here is recalculated in your browser.

Inputs used

Coefficient A (of x²): 4
Coefficient B (of xy): 3
Coefficient C (of y²): 9

Results

Conic type: Ellipse
Discriminant B² − 4AC: -135.0000
Axis rotation angle: 74.5181 °
A + C: 13.0000

Step by step

1Values used
2Discriminant test
3Axis rotation
4Conic type= Ellipse
5Discriminant B² − 4AC= -135.0000
6Axis rotation angle= 74.5181 °
7A + C= 13.0000

Assumptions and limits

The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
Figures are rounded for display to 4 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

Why does B ≠ 0 matter?

A non-zero xy term means the conic's axes are rotated relative to the coordinate axes; the reported angle removes that rotation.

Do D, E and F change the type?

No. They only translate the curve, though they can make it degenerate into a point, a line or nothing at all.