Circle Equation Forms
A circle's equation can be written in two primary forms: the standard form and the general form. Converting between them is a common task in analytic geometry and precalculus.
The Two Forms
Standard Form
Clearly shows the center (h, k) and radius r of the circle.
(x - h)² + (y - k)² = r²
General Form
Expanded polynomial form with D = -2h, E = -2k, F = h² + k² - r².
x² + y² + Dx + Ey + F = 0
Conversion Process
Expand the squared binomials, combine like terms, and move the constant to get zero on one side.
Expand, simplify, rearrange
How to Convert
- Start with (x - h)² + (y - k)² = r²
- Expand (x - h)² = x² - 2hx + h²
- Expand (y - k)² = y² - 2ky + k²
- Combine: x² - 2hx + h² + y² - 2ky + k² = r²
- Rearrange: x² + y² - 2hx - 2ky + (h² + k² - r²) = 0
Identifying D, E, and F
- D = -2h (coefficient of x)
- E = -2k (coefficient of y)
- F = h² + k² - r² (constant term)