1. What is the Circle Equation Standard Form?

The standard form of a circle equation is \( x^2 + y^2 + 2gx + 2fy + c = 0 \). This form allows us to easily identify the center and radius of the circle using specific formulas.

2. How Does the Calculator Work?

The calculator uses the formulas:

Where:

• \( g \) — Coefficient of x term divided by 2
• \( f \) — Coefficient of y term divided by 2
• \( c \) — Constant term in the equation

Explanation: These formulas are derived from completing the square for both x and y variables in the circle equation.

3. Importance of Identifying Center and Radius

Details: Knowing the center and radius of a circle is fundamental in coordinate geometry. It helps in graphing circles, solving geometric problems, and understanding the properties of circles in various applications.

4. Using the Calculator

Tips: Enter the coefficients g and f, and the constant c from the equation \( x^2 + y^2 + 2gx + 2fy + c = 0 \). The calculator will compute the center coordinates and radius.

5. Frequently Asked Questions (FAQ)

Q1: What if the radius squared is negative?
A: If \( g^2 + f^2 - c < 0 \), the equation does not represent a real circle. This indicates an imaginary circle or error in the equation.

Q2: How is this related to the standard form (x-h)² + (y-k)² = r²?
A: By expanding \( (x-h)^2 + (y-k)^2 = r^2 \), we get \( x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0 \), which matches our form with g = -h, f = -k, and c = h² + k² - r².

Q3: Can this calculator handle decimal values?
A: Yes, the calculator accepts decimal values for g, f, and c parameters.

Q4: What if my equation has different coefficients?
A: Make sure your equation is in the exact form \( x^2 + y^2 + 2gx + 2fy + c = 0 \). If coefficients of x² and y² are not 1, divide the entire equation by that coefficient first.

Q5: How accurate are the results?
A: Results are accurate to 4 decimal places, which is sufficient for most practical applications.