Circle General Form to Standard Calculator Desmos

Circle Equation Conversion:

\[ \text{General Form: } Ax^2 + Ay^2 + Dx + Ey + F = 0 \] \[ \text{Standard Form: } (x - h)^2 + (y - k)^2 = r^2 \]

Standard Form:

Desmos Formula:

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1. What is Circle General to Standard Form Conversion?

The calculator converts a circle equation from general form (Ax² + Ay² + Dx + Ey + F = 0) to standard form ((x-h)² + (y-k)² = r²). This reveals the circle's center (h,k) and radius r directly.

2. How Does the Calculator Work?

The calculator completes the square for both x and y terms:

\[ \text{1. Divide all terms by A} \] \[ \text{2. Complete the square for x and y terms} \] \[ \text{3. Rearrange into standard form} \]

Key Formulas:

Center: \( h = -\frac{D}{2A}, k = -\frac{E}{2A} \)
Radius: \( r = \sqrt{\frac{D^2 + E^2 - 4AF}{4A^2}} \)

3. Importance of Standard Form

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

4. Using the Calculator

Tips: Enter coefficients A, D, E, F from the general form equation. A cannot be zero. The Desmos formula can be copied directly into Desmos graphing calculator.

5. Frequently Asked Questions (FAQ)

Q1: What if I get a negative radius squared?
A: This means the equation doesn't represent a real circle (it might be a single point or have no real solutions).

Q2: Why must A be non-zero?
A: If A were zero, it wouldn't be a quadratic equation in both x and y, and thus wouldn't represent a circle.

Q3: Can I use this for ellipses?
A: No, this calculator is specific to circles where x² and y² coefficients are equal.

Q4: How accurate are the results?
A: Results are calculated to high precision but displayed with 2 decimal places for readability.

Q5: What if my equation has different coefficients for x² and y²?
A: That would represent an ellipse, not a circle. This calculator only works for true circles.