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Conic Sections Calculator

Identify and analyze a conic section from its general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0.

Reviewed by Christopher FloiedPublished Updated

This free online conic sections calculator provides instant results with no signup required. All calculations run directly in your browser — your data is never sent to a server. Enter your values below and see results update in real time as you type. Perfect for everyday calculations, homework, or professional use.

Results

Discriminant (B² - 4AC)

-4

Center x

0

Center y

0

Conic Type

Circle

How to Use This Calculator

1

Enter your input values

Fill in all required input fields for the Conic Sections Calculator. Most fields include unit selectors so you can work in your preferred unit system — metric or imperial, whichever matches your problem.

2

Review your inputs

Double-check that all values are correct and that you have selected the right units for each field. Incorrect units are the most common source of calculation errors and can produce results that are off by factors of 2, 10, or more.

3

Read the results

The Conic Sections Calculator instantly computes the output and displays results with units clearly labeled. All calculations happen in your browser — no loading time and no data sent to a server.

4

Explore parameter sensitivity

Try adjusting individual input values to see how the output changes. This is a quick and effective way to develop intuition about how different parameters influence the result and to identify which inputs have the largest effect.

When to Use This Calculator

  • •Use the Conic Sections Calculator when you need a quick mathematical result without writing out all the steps manually, saving time on repetitive calculations.
  • •Use it to verify hand calculations on tests or assignments and catch arithmetic mistakes.
  • •Use it when teaching or explaining mathematical concepts to others, demonstrating how changing inputs affects the result.
  • •Use it to explore the behavior of mathematical functions across a range of inputs.

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About Conic Sections Calculator

Conic sections are the curves obtained by intersecting a cone with a plane at various angles. The four types — circle, ellipse, parabola, and hyperbola — can all be described by the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. The discriminant B² - 4AC determines which type of conic the equation represents: negative for an ellipse (or circle), zero for a parabola, and positive for a hyperbola. Conic sections have been studied since ancient Greece and remain central to mathematics, physics, and engineering. Planetary orbits are ellipses, parabolas describe projectile paths in uniform gravity, hyperbolas arise in radar and GPS navigation, and circles are everywhere. Satellite dish reflectors, telescope mirrors, and architectural arches all use conic section curves. This calculator takes the six coefficients of the general equation, computes the discriminant to classify the conic, and finds the center for ellipses and hyperbolas. Understanding conic sections unifies the study of these curves under a single algebraic framework.

The Math Behind It

Every conic section satisfies the general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0. The discriminant Δ = B² - 4AC classifies the conic: Δ < 0 gives an ellipse (a circle when A = C and B = 0), Δ = 0 gives a parabola, and Δ > 0 gives a hyperbola. Degenerate cases are possible: the equation may represent a point, a line, two lines, or be inconsistent (no real solution). When B ≠ 0, the axes of the conic are rotated relative to the coordinate axes. The rotation angle θ that eliminates the xy term satisfies cot(2θ) = (A - C)/B. After rotation, the equation takes a standard form. For non-degenerate central conics (ellipses and hyperbolas), the center is found by solving the system ∂f/∂x = 0 and ∂f/∂y = 0, which gives the center at ((BE - 2CD)/(4AC - B²), (BD - 2AE)/(4AC - B²)). The eccentricity e unifies the classification: e = 0 for a circle, 0 < e < 1 for an ellipse, e = 1 for a parabola, and e > 1 for a hyperbola. All conics can also be defined as the locus of points whose distance to a focus divided by distance to a directrix equals the eccentricity, providing a unified focus-directrix definition.

Formula Reference

General Conic Equation

Ax² + Bxy + Cy² + Dx + Ey + F = 0

Variables: A, B, C, D, E, F = coefficients; B² - 4AC determines the type

Classification

B² - 4AC < 0: ellipse (or circle if A=C, B=0); = 0: parabola; > 0: hyperbola

Variables: The discriminant classifies the conic type

Worked Examples

Example 1: Identify a circle

Classify x² + y² - 4 = 0 (A=1, B=0, C=1, D=0, E=0, F=-4).

Step 1:Discriminant = 0² - 4(1)(1) = -4
Step 2:Since Δ < 0 and A = C and B = 0, this is a circle
Step 3:Center x = (0 - 0)/(4 - 0) = 0
Step 4:Center y = (0 - 0)/(4 - 0) = 0
Step 5:Radius = sqrt(4) = 2

This is a circle centered at (0, 0) with radius 2.

Example 2: Identify a hyperbola

Classify x² - y² - 1 = 0 (A=1, B=0, C=-1, D=0, E=0, F=-1).

Step 1:Discriminant = 0 - 4(1)(-1) = 4
Step 2:Since Δ > 0, this is a hyperbola
Step 3:Center x = 0, Center y = 0

This is a hyperbola centered at the origin: x² - y² = 1.

Common Mistakes & Tips

  • !Confusing the discriminant B² - 4AC for conics with the quadratic discriminant b² - 4ac — they use different coefficients.
  • !Forgetting that B = 0 in most standard form problems — the xy term only appears for rotated conics.
  • !Neglecting degenerate cases: the equation might represent a point, line(s), or have no real solution.
  • !Assuming A and C must be positive — negative values are valid and affect the classification.

Related Concepts

Parabola

A conic section with eccentricity exactly 1.

Equation of a Sphere

The 3D analog of a circle, extending conics to higher dimensions.

Latus Rectum

A key chord of any conic section through the focus.

Used in These Calculators

Calculators that build on or apply the concepts from this page:

Catenary Curve Calculator

Equation of a Sphere Calculator

Latus Rectum Calculator

Parabola Calculator

Frequently Asked Questions

How does the discriminant classify conics?

B² - 4AC < 0 means ellipse (circle if A = C, B = 0); B² - 4AC = 0 means parabola; B² - 4AC > 0 means hyperbola. This is for non-degenerate cases.

What is a degenerate conic?

A degenerate conic occurs when the equation factors or represents special cases: a single point (degenerate ellipse), one or two intersecting lines (degenerate hyperbola), or two parallel lines (degenerate parabola).

What does the xy term mean?

A nonzero B coefficient means the conic's axes are rotated relative to the coordinate axes. The rotation angle needed to eliminate the xy term is given by cot(2θ) = (A - C)/B.

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