# Equation of Circle/Cartesian

## Theorem

### Formulation 1

The equation of a circle embedded in the Cartesian plane with radius $R$ and center $\tuple {a, b}$ can be expressed as:

$\paren {x - a}^2 + \paren {y - b}^2 = R^2$

### Formulation 2

The equation:

$A \paren {x^2 + y^2} + B x + C y + D = 0$

is the equation of a circle with radius $R$ and center $\tuple {a, b}$, where:

$R = \dfrac 1 {2 A} \sqrt {B^2 + C^2 - 4 A D}$
$\tuple {a, b} = \tuple {\dfrac {-B} {2 A}, \dfrac {-C} {2 A} }$

provided:

$A > 0$
$B^2 + C^2 \ge 4 A D$

### Formulation 3

The equation of a circle with radius $R$ and center $\tuple {a, b}$ embedded in the Cartesian plane can be expressed as:

$x^2 + y^2 - 2 a x - 2 b y + c = 0$

where:

$c = a^2 + b^2 - R^2$