1. What is the General Conic Form?

The general conic form equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) represents all possible conic sections (circles, ellipses, parabolas, and hyperbolas) in a 2D plane. The coefficients determine the specific type and properties of the conic section.

2. How Does the Calculator Work?

The calculator uses the discriminant of the conic equation:

The conic type is determined as follows:

• Discriminant < 0: Ellipse (or circle if A = C and B = 0)
• Discriminant = 0: Parabola
• Discriminant > 0: Hyperbola

3. Importance of Conic Classification

Details: Identifying the conic type is essential in geometry, physics, and engineering applications. Each conic has distinct properties and applications in real-world problems.

4. Using the Calculator

Tips: Enter all six coefficients (A through F) of your conic equation. The calculator will instantly determine the conic type based on the discriminant value.

5. Frequently Asked Questions (FAQ)

Q1: What if B = 0 in the equation?
A: When B = 0, the conic is axis-aligned (not rotated). The equation simplifies and the discriminant becomes -4AC.

Q2: How is a circle different from an ellipse?
A: A circle is a special case of an ellipse where A = C and B = 0, resulting in equal major and minor axes.

Q3: Can this calculator handle degenerate cases?
A: No, this calculator only identifies non-degenerate conic sections. Degenerate cases (like two intersecting lines) require additional analysis.

Q4: What does the xy term (B coefficient) represent?
A: The Bxy term indicates rotation of the conic section relative to the coordinate axes. When B ≠ 0, the conic is rotated.

Q5: How are parabolas identified?
A: Parabolas occur when the discriminant equals exactly zero (B² - 4AC = 0).