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.
General form: Ax² + Bxy + Cy² + Dx + Ey + F = 0
1x² +0xy +1y² -4x -6y +9 = 0
What Is the Conic Section Analyzer | Circle, Ellipse, Parabola & Hyperbola?
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.
Formula
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))
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
Enter the six coefficients A through F. For a circle x²+y²=9 use A=1, B=0, C=1, F=−9. For a parabola y=x² use A=1, B=0, C=0, E=−1, F=0. Click Analyze to identify the type and compute center, foci, vertices, eccentricity, and other properties.
Example Calculation
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
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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