Standard to General Form Circle Calculator
Convert circle equations from standard form to general form with step-by-step solutions
Convert Standard Form to General Form
Standard Form
(x - h)² + (y - k)² = C
Where (h, k) is the center and r = √C is the radius
X-coordinate of circle center
Y-coordinate of circle center
Must be positive (C = r²)
Your Standard Form Equation
(x - h)² + (y - k)² = C
Example Calculation
Convert: (x - 3)² + (y + 2)² = 25
Step 1: Identify h = 3, k = -2, C = 25
Step 2: Calculate coefficients:
• D = -2h = -2(3) = -6
• E = -2k = -2(-2) = 4
• F = h² + k² - C = 9 + 4 - 25 = -12
Step 3: General form: x² + y² - 6x + 4y - 12 = 0
Conversion Formulas
From Standard Form
(x - h)² + (y - k)² = C
To General Form
x² + y² + Dx + Ey + F = 0
Where:
D = -2h
E = -2k
F = h² + k² - C
Circle Properties
Quick Tips
C must be positive for a valid circle
Standard form directly shows center and radius
General form is useful for algebraic operations
Both forms represent the same circle
Understanding the Conversion
Standard Form
The standard form (x - h)² + (y - k)² = C clearly shows the circle's center (h, k) and radius r = √C. This form is ideal for understanding the geometric properties of the circle.
General Form
The general form x² + y² + Dx + Ey + F = 0 is the expanded version where all terms are moved to one side. This form is useful for algebraic manipulations and solving systems of equations.
Conversion Process
Step 1: Expand
Expand (x-h)² and (y-k)² terms
Step 2: Collect
Group like terms together
Step 3: Rearrange
Move all terms to left side to equal zero
Mathematical Relationship
Standard Form Components:
- • (h, k): Center coordinates
- • C: Square of radius (C = r²)
- • Must have C > 0 for real circle
General Form Coefficients:
- • D = -2h (x coefficient)
- • E = -2k (y coefficient)
- • F = h² + k² - C (constant term)