Geometry

Conic Section Analyzer

Circle, Ellipse, Parabola & Hyperbola

Analyze any conic section from the general equation Ax²+Bxy+Cy²+Dx+Ey+F=0. Identifies the type, finds center, foci, vertices, asymptotes, and eccentricity, handles rotation when B≠0, and shows full discriminant-based step-by-step classification.

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General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0

1x² +0xy +1y² -4x -6y +9 = 0

Overview

What Is the Conic Section Analyzer?

A conic section is the curve formed by slicing a double cone with a plane. All conic sections satisfy a second-degree equation Ax²+Bxy+Cy²+Dx+Ey+F=0. The sign of the discriminant B²−4AC determines the type. When B≠0 (cross term present) the axis is rotated, so the calculator eliminates the cross term first by rotating coordinates before classifying and computing properties.

How it works

Conic Section Analyzer Formula and Method

General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0

Discriminant: Δ = B² − 4AC

Δ < 0 → Ellipse (or Circle if A=C, B=0)

Δ = 0 → Parabola

Δ > 0 → Hyperbola

B ≠ 0 → Rotation by θ = ½ · arctan(B / (A−C))

Instructions

How to Use

1
Enter coefficients A, B, C, D, E, F for the general conic equation
2
Use a preset (Circle, Ellipse, Parabola, Hyperbola, Rotated) for examples
3
Click Analyze to classify the conic and compute all properties
4
Read the discriminant value and classification with full step-by-step working
5
View center, foci, vertices, asymptotes (hyperbola), and eccentricity

Real example

Conic Section Analyzer Example

Example: Ellipse x²/9 + y²/4 = 1

Standard form: 4x² + 9y² − 36 = 0 → A=4, B=0, C=9, F=−36

Δ = 0² − 4·4·9 = −144 < 0 → Ellipse

Center: (0, 0) Semi-major a = 3 Semi-minor b = 2

Foci: (±√5, 0) ≈ (±2.236, 0)   Eccentricity: √5/3 ≈ 0.745

Answers

Frequently Asked Questions

What is a degenerate conic?

A degenerate conic results when the equation factors into a product of linear terms, giving a point, a line, or a pair of lines instead of a proper curve. This happens when the determinant of the 3×3 matrix [[A,B/2,D/2],[B/2,C,E/2],[D/2,E/2,F]] equals zero.

How does the calculator handle a rotated conic?

When B≠0, the cross-term Bxy means the axis is tilted. The calculator computes the rotation angle θ = ½·arctan(B/(A−C)), applies the rotation x = x'cosθ − y'sinθ, y = x'sinθ + y'cosθ to eliminate the B term, then analyzes the resulting axis-aligned equation.

What is eccentricity?

Eccentricity e measures how "stretched" a conic is. A circle has e=0, an ellipse has 0<e<1, a parabola has e=1, and a hyperbola has e>1. It equals the ratio of the distance from a point on the conic to a focus, divided by the distance from that point to the nearest directrix.

What are the asymptotes of a hyperbola?

For a hyperbola x²/a²−y²/b²=1, the asymptotes are y = ±(b/a)x. The hyperbola approaches these lines but never crosses them. The asymptotes pass through the center and have slopes ±b/a for a horizontal hyperbola.

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