Conic Sections Calculator – Guide & Formulas

Free online **conic sections calculator** — identify and analyze **circles, ellipses, parabolas, and hyperbolas** from their equations. Our **conic section formula calculator** helps you **find foci, vertices, and eccentricity** with step-by-step results. Whether you need an **ellipse calculator** or a **hyperbola calculator**, this tool provides accurate results with detailed explanations.

How to Use the Conic Sections Calculator

1. Enter the coefficients A, B, C, D, E, and F from the general conic equation.
2. The calculator identifies the conic type automatically.
3. Review the standard form, key properties (foci, vertices, axes), and eccentricity.
4. View the step-by-step conversion from general to standard form.

The Formula

Variable Definitions

• A, C: Coefficients of x² and y² terms; their relationship determines the conic type
• B: Coefficient of xy cross-term; B=0 means axes are aligned with coordinate axes
• D, E: Linear coefficients affecting center/vertex position
• F: Constant term
• e: Eccentricity: e=0 (circle), 0<e<1 (ellipse), e=1 (parabola), e>1 (hyperbola)

Example: Identifying x²/4 + y²/9 = 1

Determine the conic type and find its properties.

1. Step 1: Identify the equation form: x²/4 + y²/9 = 1. Both terms are positive with different denominators.
2. Step 2: This is an ellipse since both x² and y² terms are positive with different coefficients.
3. Step 3: Since 9 > 4, the major axis is along the y-axis. a² = 9, b² = 4.
4. Step 4: Vertices: (0, ±3). Co-vertices: (±2, 0).
5. Step 5: Foci: c² = a² — b² = 9 — 4 = 5, so c = √5. Foci at (0, ±√5).

Frequently Asked Questions