1. What Are Conic Sections?

Conic sections are curves obtained by intersecting a plane with a double-napped cone. They include circles, ellipses, parabolas, and hyperbolas, and are represented by the general quadratic equation: ax² + bxy + cy² + dx + ey + f = 0.

2. How Does The Calculator Work?

The calculator uses the discriminant (B² - 4AC) of the general conic equation to determine the type of conic section:

Where:

• If \( \Delta < 0 \): Ellipse or Circle
• If \( \Delta = 0 \): Parabola
• If \( \Delta > 0 \): Hyperbola

Note: A circle is a special case of an ellipse where a = c and b = 0.

3. Types Of Conic Sections

Circle: All points equidistant from a center point.
Ellipse: Set of points where sum of distances to two foci is constant.
Parabola: Set of points equidistant from a focus and a directrix.
Hyperbola: Set of points where difference of distances to two foci is constant.

4. Using The Calculator

Instructions: Enter the coefficients (a, b, c, d, e, f) from your conic section equation. The calculator will determine the type of conic section represented by these coefficients.

5. Frequently Asked Questions (FAQ)

Q1: What if my equation has missing terms?
A: Simply enter 0 for any missing coefficients in the general equation.

Q2: Can this calculator solve degenerate conics?
A: This calculator focuses on non-degenerate conic sections. Degenerate cases (like two intersecting lines) may not be accurately classified.

Q3: How accurate is the discriminant method?
A: The discriminant accurately classifies non-degenerate conic sections based on the quadratic form of the equation.

Q4: What about rotated conic sections?
A: The calculator works for both rotated and non-rotated conic sections as the discriminant method applies to all quadratic equations.

Q5: Can I find specific properties like foci or vertices?
A: This calculator currently only classifies the conic type. Additional calculations would be needed to find specific properties.