Conic Section Analyzer
Classify and graph a conic from its equation
About this calculator
The Conic Section Analyzer classifies any general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 as a circle, ellipse, parabola, hyperbola, or a degenerate point or line pair. It uses the discriminant B² − 4AC and the conic matrix determinant, then reports the center, rotation, semi-axes, eccentricity, foci, vertices, focus and directrix, and graphs the curve.
How to use the Conic Section Analyzer calculator
- Enter the measurements you know into the input fields.
- Read the computed properties — they update instantly as you type.
- Compare the diagram with your figure to confirm the setup is right.
- Copy the page URL to share the exact calculation.
Common examples
- x² + y² − 25 = 0 → circle, center (0, 0), radius 5
- 9x² + 4y² − 36 = 0 → ellipse, semi-axes a = 3, b = 2, eccentricity ≈ 0.7454
- x² − y = 0 (y = x²) → parabola, vertex (0, 0), focus (0, 0.25), directrix y = −0.25
- xy = 1 → hyperbola rotated 45°, eccentricity √2 ≈ 1.4142, vertices (1, 1) and (−1, −1)
- x² − y² = 0 → degenerate: two intersecting lines through the origin
Frequently asked questions
How does the Conic Section Analyzer calculator work?
Enter the known measurements of your figure and the calculator applies the standard geometric formulas to find the remaining properties, alongside a diagram that reflects your numbers. All values are computed with high-precision arithmetic.
When would I use the Conic Section Analyzer calculator?
It helps with geometry homework, technical drawing, construction and DIY layout, and any situation where you know some measurements of a shape and need the rest.
How accurate are the results?
Formulas are evaluated at 30 significant digits of precision internally, including roots and trigonometric functions. Displayed values are rounded for readability, but the underlying computation does not accumulate floating-point drift.
Can I share a specific calculation with someone else?
Yes. Every value you enter updates the URL, so copying the address bar and sharing the link reopens the exact same figure and results for whoever clicks it.