Enter the general quadratic equation
Use coefficients from Ax² + Bxy + Cy² + Dx + Ey + F = 0.
Plotly graph
The plot traces the implicit curve where Ax² + Bxy + Cy² + Dx + Ey + F equals zero.
Submit the form to draw the current conic.
Example data table
| Example | Equation | Coefficients (A, B, C, D, E, F) | Expected type |
|---|---|---|---|
| 1 | x² + y² - 4x - 6y + 9 = 0 | (1, 0, 1, -4, -6, 9) | Circle |
| 2 | 9x² + 4y² - 36 = 0 | (9, 0, 4, 0, 0, -36) | Ellipse |
| 3 | y² - 8x = 0 | (0, 0, 1, -8, 0, 0) | Parabola |
| 4 | x² - y² - 1 = 0 | (1, 0, -1, 0, 0, -1) | Hyperbola |
| 5 | x² - 2x + 1 = 0 | (1, 0, 0, -2, 0, 1) | Degenerate pair or repeated line |
Formula used
1) General equation
Every input is interpreted as:
Ax² + Bxy + Cy² + Dx + Ey + F = 0
This single model covers circles, ellipses, parabolas, hyperbolas, and degenerate cases.
2) Discriminant test
Δ = B² - 4AC
If Δ < 0, the conic is ellipse-like. If Δ = 0, it is parabola-like. If Δ > 0, it is hyperbola-like.
3) Degeneracy test
The calculator evaluates the augmented determinant:
| A B/2 D/2 |
| B/2 C E/2 |
| D/2 E/2 F |
A zero determinant signals a degenerate conic such as intersecting lines, parallel lines, or a single point.
4) Rotation angle
tan(2θ) = B / (A - C)
A nonzero xy term means the conic is tilted. The calculator rotates axes to reveal the standard form.
5) Center formulas
When a center exists:
x₀ = (BE - 2CD) / (4AC - B²)
y₀ = (BD - 2AE) / (4AC - B²)
These formulas locate the center of circles, ellipses, and hyperbolas.
6) Canonical metrics
The rotated quadratic matrix gives principal coefficients λ₁ and λ₂.
From them, the calculator derives semi-axes, radius, eccentricity, focal distance, and canonical descriptions whenever the curve is real.
How to use this calculator
- Enter the six coefficients from your quadratic equation.
- Keep graph bounds wide enough to capture the visible curve.
- Press Identify conic to classify the shape.
- Review the conic type, discriminant, center or vertex, rotation, and canonical form.
- Inspect the graph to confirm the geometry visually.
- Export the analysis as CSV or PDF for notes, homework, or reports.
FAQs
1) What equation format does the calculator accept?
It accepts the full second-degree form Ax² + Bxy + Cy² + Dx + Ey + F = 0. Enter zero for any missing term.
2) How does it tell a circle from an ellipse?
A circle occurs when the squared terms are equal and the xy term vanishes after simplification. Otherwise, an ellipse generally has unequal principal axes.
3) Why is the Bxy term important?
That term indicates rotation. The calculator removes it with an axis rotation, then identifies the conic from the simplified principal coefficients.
4) What is a degenerate conic?
A degenerate conic does not form a usual curve. It may collapse into intersecting lines, parallel lines, a repeated line, or a single point.
5) Does the calculator handle rotated parabolas?
Yes. It computes the rotation angle automatically, then extracts the vertex, focus information, and axis direction in rotated coordinates.
6) Why might the graph look empty?
The curve may lie outside your graph window, the range may be too narrow, or the equation may represent an imaginary or degenerate case.
7) Can I use decimal coefficients?
Yes. Decimal values work well. The calculator also rounds displayed results using the precision setting you choose.
8) What do the CSV and PDF exports contain?
They include the entered equation, classification, discriminant, determinant, key geometric values, and the canonical summary shown in the results panel.