Conic Sections Calculator | Identify Ellipse, Parabola, Hyperbola & Circle

Conic Sections Calculator

Identify the type of conic section from its general equation

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

Use Preset Examples

Examples

Circle: x² + y² - 25 = 0 (A=1, B=0, C=1, D=0, E=0, F=-25)
Ellipse: 4x² + 9y² - 36 = 0 (A=4, B=0, C=9, D=0, E=0, F=-36)
Parabola: x² - 4y = 0 (A=1, B=0, C=0, D=0, E=-4, F=0)
Hyperbola: x² - y² - 1 = 0 (A=1, B=0, C=-1, D=0, E=0, F=-1)

Classification Rule

The discriminant B² - 4AC determines the type: < 0 (ellipse/circle), = 0 (parabola), > 0 (hyperbola).

Understanding Conic Sections Calculator: A Comprehensive Guide

Conic sections are curves formed by intersecting a plane with a cone.
The general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 represents all conics.
The discriminant B² - 4AC determines the type of conic section.

Conic sections are fundamental curves in mathematics, formed when a plane intersects a cone at different angles. The four main types are circles, ellipses, parabolas, and hyperbolas.

The general second-degree equation Ax² + Bxy + Cy² + Dx + Ey + F = 0 can represent any conic section, depending on the values of the coefficients.

The discriminant Δ = B² - 4AC is the key to classification: Δ < 0 indicates an ellipse (or circle if A = C and B = 0), Δ = 0 indicates a parabola, and Δ > 0 indicates a hyperbola.

Basic Classification Examples

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

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

Enter the coefficients A, B, C, D, E, F from your equation.
Use preset examples to understand different conic types.
Interpret the discriminant and classification result.

To use the calculator, identify the coefficients from your equation in the form Ax² + Bxy + Cy² + Dx + Ey + F = 0. Enter each coefficient in the corresponding field.

Input Guidelines:

At least one of A, B, or C must be non-zero for a valid conic section.

Use the preset dropdown to see examples of each conic type with their coefficients.

Missing terms have coefficient 0 (e.g., if there's no xy term, then B = 0).

Understanding Results:

The calculator uses the discriminant B² - 4AC to classify the conic.

Additional conditions distinguish circles from ellipses and identify degenerate cases.

Usage Examples

For x² + y² - 25 = 0: A=1, B=0, C=1, D=0, E=0, F=-25 → Circle
For 4x² + 9y² - 36 = 0: A=4, B=0, C=9, D=0, E=0, F=-36 → Ellipse
For y² - 4x = 0: A=0, B=0, C=1, D=-4, E=0, F=0 → Parabola

Real-World Applications of Conic Sections Calculations

Engineering and architecture use conic sections in design.
Physics applications include planetary orbits and projectile motion.
Optics and acoustics rely on reflective properties of conics.

Conic sections appear throughout science and engineering, making their identification and analysis crucial for many applications.

Astronomy and Physics:

Planetary orbits are ellipses with the sun at one focus (Kepler's laws).

Projectile paths under gravity follow parabolic trajectories.

Hyperbolic paths describe certain spacecraft trajectories and particle interactions.

Engineering and Architecture:

Parabolic reflectors focus radio waves and light (satellite dishes, telescopes).

Elliptical arches distribute weight efficiently in construction.

Hyperbolic cooling towers optimize airflow and structural stability.

Real-World Examples

Satellite dish design uses parabolic curves for signal focusing.
Bridge arches often use elliptical or parabolic shapes for strength.
Planetary motion follows elliptical orbits around stars.

Common Misconceptions and Correct Methods in Conic Sections

Students often confuse the classification criteria.
The orientation and position don't affect the basic type.
Degenerate cases require special consideration.

Many students struggle with conic section classification, often due to misconceptions about the discriminant or confusion between different forms of equations.

Misconception 1: Only Standard Forms

Any second-degree equation can represent a conic, not just standard forms like x²/a² + y²/b² = 1.

Misconception 2: Position Affects Type

Translation and rotation don't change the fundamental type of conic section.

Misconception 3: Ignoring the Discriminant

The discriminant B² - 4AC is the most reliable method for classification.

Correct Method: Use the General Equation

Always start with the general form and calculate the discriminant for reliable classification.

Misconceptions & Corrections

x² + y² + 4x - 6y + 9 = 0 is still a circle despite the linear terms.
2x² + 3y² - 4x + 6y - 1 = 0 is an ellipse (discriminant = -24 < 0).
x² - 2xy + y² + 3x - y = 0 is a parabola (discriminant = 0).

Mathematical Derivation and Examples

Derivation of the discriminant classification rule.
Worked examples for each type of conic section.
Special cases and degenerate conics.

The classification of conics using the discriminant comes from the theory of quadratic forms. When we diagonalize the quadratic part Ax² + Bxy + Cy², the eigenvalues determine the type.

For the general equation Ax² + Bxy + Cy² + Dx + Ey + F = 0, the discriminant Δ = B² - 4AC tells us about the nature of the conic:

• If Δ < 0: The conic is an ellipse (or circle if A = C and B = 0)

• If Δ = 0: The conic is a parabola

• If Δ > 0: The conic is a hyperbola

Example: For x² + 4xy + 4y² - 6x + 8y + 1 = 0, we have A=1, B=4, C=4, so Δ = 16 - 16 = 0, indicating a parabola.

Mathematical Examples

Classification formula: Δ = B² - 4AC
Circle: x² + y² - 16 = 0 (Δ = 0 - 4(1)(1) = -4 < 0, A = C)
Ellipse: 9x² + 4y² = 36 (Δ = 0 - 4(9)(4) = -144 < 0, A ≠ C)
Parabola: x² + 2xy + y² = 1 (Δ = 4 - 4(1)(1) = 0)
Hyperbola: x² - y² = 1 (Δ = 0 - 4(1)(-1) = 4 > 0)