1. What Is The Conic Equation?

The general conic equation \( ax^2 + bxy + cy^2 + dx + ey + f = 0 \) represents all conic sections: circles, ellipses, parabolas, and hyperbolas. The type of conic is determined by the discriminant \( b^2 - 4ac \).

2. How Does The Calculator Work?

The calculator analyzes the discriminant and coefficients to determine the conic type:

Where:

• \( \Delta < 0 \): Ellipse (or circle if \( a = c \) and \( b = 0 \))
• \( \Delta = 0 \): Parabola
• \( \Delta > 0 \): Hyperbola

Explanation: The calculator also computes the center coordinates for non-parabolic conics using the formula: \[ x_c = \frac{be - 2cd}{4ac - b^2}, \quad y_c = \frac{bd - 2ae}{4ac - b^2} \]

3. Importance Of Conic Equations

Details: Conic sections have wide applications in physics, engineering, astronomy, and many other fields. They describe orbits, reflectors, antenna designs, and architectural structures.

4. Using The Calculator

Tips: Enter all six coefficients (a through f) as real numbers. The calculator will determine the conic type and provide its center coordinates when applicable.

5. Frequently Asked Questions (FAQ)

Q1: What if the discriminant is exactly zero?
A: This indicates a parabola, which has no center point.

Q2: Can all conics be represented this way?
A: Yes, all conic sections can be expressed in the general second-degree equation form.

Q3: What does a zero coefficient mean?
A: Zero coefficients simplify the equation. For example, if b=0, the conic's axes are aligned with the coordinate axes.

Q4: How are degenerate cases handled?
A: The calculator may not identify degenerate cases (like two intersecting lines) which require additional analysis.

Q5: What precision does the calculator use?
A: Calculations are performed with floating-point precision and results are rounded to two decimal places for display.