1. What is the Standard Form of a Circle?

The standard form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where (h,k) is the center and r is the radius. This form clearly shows the circle's geometric properties.

2. How Does the Calculator Work?

The calculator converts from general form \(x^2 + y^2 + Dx + Ey + F = 0\) to standard form:

Where:

• \( h = -D/2 \)
• \( k = -E/2 \)
• \( r = \sqrt{h^2 + k^2 - F} \)

Explanation: The calculator completes the square algebraically to transform the equation into standard form.

3. Importance of Standard Form

Details: Standard form immediately reveals the circle's center and radius, making it essential for graphing and geometric analysis.

4. Using the Calculator

Tips: Enter the coefficients of x and y, and the constant term from your circle's equation. The calculator will output the standard form.

5. Frequently Asked Questions (FAQ)

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

Q2: Can I use this for equations with xy terms?
A: No, this calculator only works for equations in the form \(x^2 + y^2 + Dx + Ey + F = 0\).

Q3: What if my equation has coefficients on x² and y²?
A: First divide all terms by that coefficient to make them 1 before using the calculator.

Q4: How precise are the results?
A: Results are rounded to 4 decimal places for readability.

Q5: Can this calculator handle vertical or horizontal shifts?
A: Yes, the standard form precisely shows any horizontal (h) or vertical (k) shifts.