Conic Sections Calculator: Circle, Parabola, Ellipse & Hyperbola
Use multi-mode tools to go equation → features or features → equation. v1.1 supports axes-aligned conics only (no xy term).
Background
Conic sections are curves formed by intersecting a plane with a double cone. Depending on the angle of intersection, the result is a circle, ellipse, parabola, or hyperbola. Algebraically, conics appear as second-degree equations in x and y. When no xy term is present, the conic is aligned with the coordinate axes.
How to Use This Conic Sections Calculator
- Identify + Convert: Enter a general equation in the form Ax² + Cy² + Dx + Ey + F = 0 (no xy term).
- We classify the conic (circle, parabola, ellipse, hyperbola).
- We complete the square (if needed) and convert to standard form.
- Mode tools: Enter geometric features (center, radius, a, b, p) to build equations instantly.
- Turn on Graph Labels to visualize foci, directrix, or asymptotes.
- Enable Sanity Check to expand back to general form and verify correctness.
How This Calculator Works
- Detects conic type based on signs and coefficients of x² and y².
- Uses completing the square to convert general form into standard form.
- Extracts geometric features: center, vertex, focus, radius, axes lengths, asymptotes.
- Graph preview uses parametric sampling (ellipse/hyperbola) and function form (parabola).
- Optional rounding applies only to display — internal math stays precise.
Standard Forms & Formulas Used
Circle
Standard form:
Parabola
Vertical:
Horizontal:
Ellipse
Focal distance: c² = a² − b²
Hyperbola
Asymptotes: y − k = ±(b/a)(x − h)
Example Problems
Example 1 — Identify a circle
Convert x² + y² − 4x + 6y − 12 = 0 to standard form.
- Group x and y terms.
- Complete the square for both variables.
- Result: (x − 2)² + (y + 3)² = 25
- Center = (2, −3), radius = 5.
Example 2 — Ellipse from general form
Convert 4x² + 9y² − 36 = 0.
- Divide both sides by 36.
- Standard form: x²/9 + y²/4 = 1
- a = 3, b = 2 → foci at ±√5 along major axis.
Example 3 — Hyperbola
Identify x² − y² − 1 = 0.
- Rewrite: x² − y² = 1
- Opposite signs → hyperbola.
- Asymptotes: y = ±x
Frequently Asked Questions
Q: Why doesn’t this support an xy term?
An xy term introduces rotation. This version supports axes-aligned conics only. A rotated conic requires diagonalization of the quadratic form.
Q: How do I know if it’s an ellipse or hyperbola?
If x² and y² have the same sign → ellipse (or circle). If opposite signs → hyperbola.
Q: What determines parabola direction?
Which variable is squared. If x² appears (but not y²), the parabola opens up/down. If y² appears, it opens left/right.
Q: Why use the sanity check?
It expands the standard form back to general form so you can verify algebra accuracy — extremely helpful for exam prep.
Q: Is this good for SAT/AP/College Algebra exams?
Absolutely. Completing the square and identifying conics are high-frequency exam skills.
