Conic Sections Calculator

Identify and classify conic sections from their general equation

Enter the coefficients of the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 to determine whether it represents a circle, ellipse, parabola, or hyperbola using discriminant analysis.

Coefficient of x² term (default: 0)

Coefficient of xy term (default: 0)

Coefficient of y² term (default: 0)

Coefficient of x term (default: 0)

Coefficient of y term (default: 0)

Constant term (default: 0)

Example Conic Sections

Click on any example to load it into the calculator and see the classification process

Circle Example

circle

Standard circle equation: x² + y² - 25 = 0

A: 1, B: 0, C: 1

D: 0, E: 0, F: -25

Ellipse Example

ellipse

Standard ellipse equation: 4x² + 9y² - 36 = 0

A: 4, B: 0, C: 9

D: 0, E: 0, F: -36

Parabola Example

parabola

Standard parabola equation: x² - 4y = 0

A: 1, B: 0, C: 0

D: 0, E: -4, F: 0

Hyperbola Example

hyperbola

Standard hyperbola equation: x² - y² - 1 = 0

A: 1, B: 0, C: -1

D: 0, E: 0, F: -1

Other Titles
Understanding Conic Sections Calculator: A Comprehensive Guide
Learn to identify and classify conic sections using the general equation and discriminant analysis for geometry and calculus applications.

What are Conic Sections and Why Do They Matter?

  • Fundamental curves formed by intersecting a plane with a cone
  • Essential for understanding geometry, physics, and engineering applications
  • Classification using the discriminant method for any equation form
Conic sections are among the most important curves in mathematics, formed when a plane intersects a cone at different angles. These curves include circles, ellipses, parabolas, and hyperbolas, each with unique properties and real-world applications.
The Four Types of Conic Sections:
• Circle: A special case of ellipse where all points are equidistant from the center
• Ellipse: An oval-shaped curve where the sum of distances to two foci is constant
• Parabola: A curve where each point is equidistant from a focus and directrix
• Hyperbola: A curve with two branches where the difference of distances to two foci is constant
The General Equation:
All conic sections can be represented by the general second-degree equation: Ax² + Bxy + Cy² + Dx + Ey + F = 0. The coefficients determine which type of conic the equation represents.

Basic Conic Section Examples

  • Circle: x² + y² = 25 (A=1, C=1, B=0, Discriminant = -4)
  • Ellipse: 4x² + 9y² = 36 (A=4, C=9, B=0, Discriminant = -144)
  • Parabola: y² = 4x (A=0, C=1, B=0, Discriminant = 0)
  • Hyperbola: x² - y² = 1 (A=1, C=-1, B=0, Discriminant = 4)

Step-by-Step Guide to Using the Conic Sections Calculator

  • How to enter coefficients from any conic equation
  • Understanding the discriminant calculation process
  • Interpreting results and classification criteria
Using our conic sections calculator is straightforward, but understanding the process helps you learn the mathematics behind conic classification.
Step 1: Identify Your Equation Form
Start with any equation containing x² and/or y² terms. Rearrange it to the general form Ax² + Bxy + Cy² + Dx + Ey + F = 0 by moving all terms to one side.
Step 2: Extract the Coefficients
• A: coefficient of x² term (if no x² term, A = 0)
• B: coefficient of xy term (if no xy term, B = 0)
• C: coefficient of y² term (if no y² term, C = 0)
• D, E, F: coefficients of x, y, and constant terms respectively
Step 3: Enter Values and Calculate
Input your coefficients into the calculator. The discriminant Δ = B² - 4AC will be calculated automatically to determine the conic type.

Step-by-Step Classification Examples

  • For 3x² + 2y² - 12 = 0: A=3, B=0, C=2, D=0, E=0, F=-12 → Ellipse
  • For y² - 8x = 0: A=0, B=0, C=1, D=-8, E=0, F=0 → Parabola
  • For x² - 4y² + 16 = 0: A=1, B=0, C=-4, D=0, E=0, F=16 → Hyperbola

Real-World Applications of Conic Sections

  • Astronomical applications in orbital mechanics
  • Engineering and architectural design principles
  • Physics applications in optics and acoustics
Conic sections appear everywhere in science and engineering, making their identification and analysis crucial for understanding natural phenomena and designing technological systems.
Astronomy and Space Science:
• Planetary orbits follow elliptical paths with the sun at one focus (Kepler's First Law)
• Comet trajectories can be elliptical, parabolic, or hyperbolic depending on their energy
• Satellite orbits are carefully designed ellipses to maintain proper coverage and altitude
Engineering and Technology:
• Parabolic reflectors in satellite dishes focus radio signals to a single point
• Elliptical gears provide non-uniform motion in specialized machinery
• Hyperbolic cooling towers optimize airflow and structural stability in power plants
Architecture and Design:
• Elliptical arches distribute weight efficiently in bridge and building construction
• Parabolic arches create strong, aesthetically pleasing architectural features

Real-World Conic Applications

  • GPS satellites use elliptical orbits calculated using conic section mathematics
  • The Hubble Space Telescope uses a parabolic primary mirror for perfect focus
  • Nuclear power plant cooling towers use hyperbolic shapes for optimal airflow

Common Misconceptions and Correct Classification Methods

  • Why visual appearance can be misleading for classification
  • The importance of the discriminant over intuitive methods
  • Handling special cases and degenerate conics properly
Students often make classification errors when learning about conic sections. Understanding these common misconceptions helps ensure accurate identification.
Misconception 1: Visual Classification
Many students try to classify conics by their appearance when graphed. However, the equation's coefficients provide the only reliable classification method, regardless of how the curve looks when plotted.
Misconception 2: Ignoring Rotation Effects
When B ≠ 0 (xy term present), the conic is rotated from standard position. The type is still determined by the discriminant, not by the apparent orientation of the curve.
Misconception 3: Position Affects Type
The D and E coefficients only affect the position (translation) of the conic, not its fundamental type. A circle remains a circle regardless of where it's centered.
Correct Method: Always Use the Discriminant
The discriminant Δ = B² - 4AC is the universal method for classification: Δ < 0 (ellipse/circle), Δ = 0 (parabola), Δ > 0 (hyperbola).

Misconception Corrections

  • x² + y² + 6x - 4y + 9 = 0 is still a circle despite the linear terms (Δ = -4)
  • xy + x - y = 0 is a hyperbola even though it looks different (Δ = 1 > 0)
  • 2x² + 4xy + 2y² - 6 = 0 is a parabola because Δ = 16 - 16 = 0

Mathematical Theory and Advanced Examples

  • Derivation of the discriminant classification rule
  • Matrix representation and eigenvalue analysis
  • Special cases including degenerate conics
The discriminant method for classifying conics comes from linear algebra and the theory of quadratic forms. Understanding this theory provides insight into why the method works universally.
Quadratic Form Analysis:
The quadratic terms Ax² + Bxy + Cy² can be written as a matrix: [x y] × [A B/2; B/2 C] × [x; y]. The eigenvalues of this matrix determine the conic type.
The Discriminant Formula:
Δ = B² - 4AC equals 4 times the determinant of the coefficient matrix. This invariant quantity remains unchanged under rotation, making it perfect for classification.
Classification Rules:
• Δ < 0: Both eigenvalues have the same sign → Ellipse (or circle if A = C and B = 0)
• Δ = 0: One eigenvalue is zero → Parabola
• Δ > 0: Eigenvalues have opposite signs → Hyperbola
Degenerate Cases:
When the equation factors into linear terms, we get degenerate conics: intersecting lines, parallel lines, or single points.

Advanced Mathematical Examples

  • Discriminant formula: Δ = B² - 4AC for any conic equation
  • Circle condition: Δ < 0 AND A = C AND B = 0
  • Rotated ellipse: 5x² + 4xy + 8y² = 36 (Δ = 16 - 160 = -144 < 0)
  • Degenerate case: x² - y² = 0 factors to (x-y)(x+y) = 0 (two intersecting lines)