1. What is the General to Standard Form Conversion?

The calculator converts the general form of a circle equation (x² + y² + Dx + Ey + F = 0) to the standard form ((x - h)² + (y - k)² = r²), which clearly shows the center (h,k) and radius r of the circle.

2. How Does the Calculator Work?

The calculator uses the following conversion formulas:

Where:

• \( D, E, F \) — Coefficients from the general form
• \( h \) — x-coordinate of center
• \( k \) — y-coordinate of center
• \( r \) — Radius of the circle

Explanation: The conversion is done by completing the square for both x and y terms in the general form equation.

3. Importance of Standard Form

Details: The standard form makes it immediately obvious where the circle is centered and what its radius is, which is essential for graphing and geometric analysis.

4. Using the Calculator

Tips: Enter the D, E, and F coefficients from your general form equation. The calculator will determine if the equation represents a valid circle (r² > 0).

5. Frequently Asked Questions (FAQ)

Q1: What if I get "Invalid circle equation"?
A: This means the radius would be imaginary (r² ≤ 0), indicating your equation doesn't represent a real circle.

Q2: Can this calculator handle fractions?
A: Yes, you can enter decimal equivalents of fractions (e.g., 0.5 for 1/2).

Q3: What's the difference between general and standard form?
A: General form shows all terms expanded, while standard form clearly displays the geometric properties of the circle.

Q4: How do I know if my general form represents a circle?
A: After conversion, if r² > 0, it's a valid circle. If r² = 0, it's a single point, and if r² < 0, it's not a real circle.

Q5: Can I convert back from standard to general form?
A: Yes, by expanding the squared terms in the standard form equation.