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Conic Section Identifier

Identify the conic section type (circle, ellipse, parabola, or hyperbola) from the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. Get step-by-step classification, key properties, standard form, and an interactive graph.

Conic Section Identifier
Examples:
GENERAL EQUATION
Ax² + Bxy + Cy² + Dx + Ey + F = 0

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About Conic Section Identifier

The Conic Section Identifier classifies any general second-degree equation of the form Ax² + Bxy + Cy² + Dx + Ey + F = 0 into one of the four conic section types: circle, ellipse, parabola, or hyperbola. It also detects degenerate cases such as points, single lines, intersecting lines, and parallel lines. Enter the six coefficients and get instant identification with a detailed step-by-step classification, key geometric properties, and an interactive graph.

The Four Conic Sections

⭕
Circle
All points equidistant from a center. Special case of ellipse where A = C and B = 0. Eccentricity = 0.
⬮
Ellipse
Oval curve with two foci. The sum of distances from any point to both foci is constant. Eccentricity between 0 and 1.
⌒
Parabola
U-shaped curve where each point is equidistant from a focus and a directrix line. Eccentricity = 1.
⟠
Hyperbola
Two mirror-image branches. The difference of distances from any point to the two foci is constant. Eccentricity > 1.

How to Identify a Conic Section

The key to identifying a conic section from its general equation is the discriminant \(\Delta = B^2 - 4AC\), calculated from the coefficients of the second-degree terms. This value is invariant under rotation of axes.

Discriminant (B² − 4AC)Conic TypeAdditional Condition
< 0EllipseA ≠ C or B ≠ 0
< 0CircleA = C and B = 0
= 0ParabolaA or C (not both) is 0
> 0Hyperbola—

The Role of the Bxy Term

When the coefficient B is non-zero, the conic's principal axes are rotated relative to the x- and y-coordinate axes. To eliminate the xy term, we rotate the axes by angle \(\theta = \frac{1}{2}\arctan\left(\frac{B}{A - C}\right)\). After rotation, the equation takes a standard form without the cross term, making it easier to identify properties like center, foci, and vertices.

Degenerate Conic Sections

Not every second-degree equation produces a full conic curve. Degenerate cases occur when the plane passes through the apex of the cone:

How to Use the Conic Section Identifier

  1. Enter coefficients: Type the values of A, B, C, D, E, and F from your general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.
  2. Use quick examples: Click a preset button (Circle, Ellipse, Parabola, Hyperbola, or Rotated) to auto-fill sample coefficients.
  3. Click Identify: Press the "Identify Conic Section" button to classify the equation.
  4. Review results: See the conic type, discriminant, geometric properties (center, foci, eccentricity, axes), step-by-step solution, and interactive graph.
  5. Explore the graph: Drag to pan, scroll to zoom, or use the +/− buttons. The graph plots the actual curve from the given equation.

Practical Applications

Conic sections appear throughout science and engineering. Planetary orbits are ellipses (Kepler's first law). Satellite dishes and car headlights use parabolic reflectors to focus signals. Hyperbolas arise in navigation systems (LORAN) and in the paths of objects with enough energy to escape a gravitational field. Circles are ubiquitous in wheels, gears, and clock faces.

FAQ

What is a conic section?
A conic section is a curve formed by the intersection of a plane and a double-napped cone. The four types are circle, ellipse, parabola, and hyperbola. They are all described by the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.
How do you identify a conic section from its equation?
Compute the discriminant Δ = B² − 4AC. If Δ < 0, it's an ellipse (or a circle if A = C and B = 0). If Δ = 0, it's a parabola. If Δ > 0, it's a hyperbola. Always check for degenerate cases too.
What is the discriminant of a conic section?
The discriminant Δ = B² − 4AC is computed from the coefficients of the second-degree terms. It determines the conic type and is invariant under rotation, meaning it gives the same value regardless of how the coordinate axes are oriented.
What is a degenerate conic?
A degenerate conic occurs when the general equation factors into simpler curves: a single point (degenerate ellipse), two intersecting lines (degenerate hyperbola), or parallel/coincident lines (degenerate parabola). These arise when the cutting plane passes through the apex of the cone.
What does the Bxy term mean in the general equation?
The Bxy term indicates that the conic's axes are rotated relative to the coordinate axes. When B ≠ 0, you eliminate the xy term by rotating axes by θ = ½ arctan(B/(A−C)), which reveals the standard form of the conic.

Reference this content, page, or tool as:

"Conic Section Identifier" at https://MiniWebtool.com/conic-section-identifier/ from MiniWebtool, https://MiniWebtool.com/

by miniwebtool team. Updated: 2026-04-02

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