Conic Sections Calculator

Enter the coefficients of Ax² + Bxy + Cy² + Dx + Ey + F = 0.

This result is an estimate only. Double-check important figures before relying on them.

How it is calculated

Discriminant = B² − 4AC
Determinant of the 3×3 matrix [A B/2 D/2; B/2 C E/2; D/2 E/2 F]
Determinant 0 → degenerate; otherwise the sign of the discriminant sets the type

For the roots of a quadratic, use the quadratic equation calculator.

Worked example

9x² + 4y² − 36 = 0:

B² − 4AC = −144, below zero
A ≠ C, so an ellipse

Why your number may differ

Equations that factor into lines, or that have no real points such as x² + y² + 1 = 0, are reported by the determinant test and may show as an ellipse. Only the type is reported.

Frequently asked questions

How do I tell the type of conic?

Look at B² − 4AC: below zero is an ellipse (a circle when A = C and B = 0), zero is a parabola, above zero is a hyperbola.

What is a degenerate conic?

One that collapses to a point, a line or two lines, such as x² − y² = 0. It shows when the determinant of the conic's matrix is zero.

Does it find the centre or foci?

No. It identifies the type only.