1. What is the General Form of a Circle Equation?

The general form of a circle equation is \( x² + y² + Dx + Ey + F = 0 \). This equation can be converted to the standard form \( (x - h)² + (y - k)² = r² \) where (h, k) is the center and r is the radius.

2. How Does the Calculator Work?

The calculator converts the general form to standard form:

Where:

• \( h = -D/2 \) — x-coordinate of center
• \( k = -E/2 \) — y-coordinate of center
• \( r = \sqrt{h² + k² - F} \) — radius of circle

Explanation: The calculator completes the square to transform the general form into the more useful standard form.

3. Importance of Circle Equations

Details: The standard form makes it easy to identify the circle's center and radius, which are essential for graphing and geometric calculations.

4. Using the Calculator

Tips: Enter the coefficients D, E, and constant F from your general form equation. The calculator will output the standard form, center, and radius.

5. Frequently Asked Questions (FAQ)

Q1: What if the radius comes out negative?
A: A negative value under the square root means the equation doesn't represent a real circle (the radius would be imaginary).

Q2: How is this different from the standard form?
A: The general form is expanded while the standard form directly shows the center and radius.

Q3: Can this represent a single point?
A: Yes, when radius = 0, the equation represents a single point (the center).

Q4: What about degenerate cases?
A: If the equation doesn't represent a real circle (radius² ≤ 0), the calculator will still show the calculated values.

Q5: Can I use this for 3D circles?
A: No, this is only for circles in 2D space (the xy-plane).