Conic Sections Calculator (identify Ax² + Bxy + Cy² + Dx + Ey + F = 0)

Q: How do I tell what conic an equation is?

Compute B² − 4AC: negative for an ellipse, zero for a parabola, positive for a hyperbola.

Q: When is it a circle?

When B = 0 and A = C (and the curve is real and non-degenerate).

Q: What is a degenerate conic?

A case where the equation gives a point, one or two lines, or no points at all.

Understanding the conic sections calculator — concept and principle

This calculator identifies the curve described by a general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0: ellipse, circle, parabola or hyperbola, including degenerate cases. It gives the discriminant, the degeneracy determinant, the rotation angle that removes the xy term, and the centre. Use it for precalculus and analytic geometry problems.

B² − 4AC < 0 and A ≠ C: an ellipse, not a circle.

How it is calculated

Discriminant B² − 4AC: negative → ellipse (a circle if B = 0 and A = C), zero → parabola, positive → hyperbola.

Δ = det[[A, B/2, D/2], [B/2, C, E/2], [D/2, E/2, F]]: if Δ = 0 the conic is degenerate (a point, one or two lines, or nothing). An ellipse with (A + C)·Δ > 0 has no real points.

Rotation angle θ with cot 2θ = (A − C) ÷ B removes the xy term.

Centre (ellipses and hyperbolas): solve 2Ax + By + D = 0 and Bx + 2Cy + E = 0.

The principle

Every second-degree curve in the plane is a slice through a double cone. Rotating the axes can remove the xy term, and shifting them can remove the linear terms, leaving a standard form. The discriminant B² − 4AC does not change under rotation, so it tells the type without doing the rotation. The 3 × 3 determinant Δ is also unchanged by rotations and shifts, and it is zero exactly when the equation factors into lines or collapses to a point.

Worked example

Default inputs: 4x² + 9y² − 16x + 18y − 11 = 0 (A = 4, B = 0, C = 9, D = −16, E = 18, F = −11).

Discriminant = 0² − 4 × 4 × 9 = −144, negative, so an ellipse; A ≠ C, so not a circle.
Δ = 4(9 × (−11) − 9²) + (−8)(0 × 9 − 9 × (−8)) = −720 − 576 = −1,296, not zero; (A + C)Δ < 0, so the ellipse is real.
B = 0, so the rotation angle is 0°.
Centre: 8x − 16 = 0 and 18y + 18 = 0 give (2, −1). Type: Ellipse; radius — (not a circle).

How to read the result

Completing the squares gives 4(x − 2)² + 9(y + 1)² = 36, i.e. (x − 2)²/9 + (y + 1)²/4 = 1: an ellipse centred at (2, −1) with semi-axes 3 and 2. The discriminant sign is the first check; Δ = 0 warns that the "curve" is really lines or a point.

Common mistakes

Using B² − AC or forgetting the factor 4.
Forgetting that B is the full xy coefficient, so B/2 goes into the matrix.
Calling every ellipse with A = C a circle when B ≠ 0.
Ignoring degenerate cases.

Keep in mind: Coefficients entered as rounded decimals can push a borderline case (such as a parabola, discriminant 0) to another type. Use exact values where possible.

Frequently asked questions

How do I tell what conic an equation is?