Equation of Circle/Cartesian/Formulation 2

Theorem

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

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:

$A > 0$

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

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

$=$

$\ds 0$

$\ds \leadstoandfrom \ \ $

$\ds x^2 + y^2 + \frac B A x + \frac C A y$

$=$

$\ds - \frac D A$

$\ds \leadstoandfrom \ \ $

$\ds x^2 + 2 \frac B {2 A} x + \frac {B^2} {4 A^2} + y^2 + 2 \frac C {2 A} y + \frac {C^2} {4 A^2}$

$=$

$\ds \frac {B^2} {4 A^2} + \frac {C^2} {4 A^2} - \frac D A$

$\ds \leadstoandfrom \ \ $

$\ds \paren {x + \frac B {2 A} }^2 + \paren {y + \frac C {2 A} }^2$

$=$

$\ds \frac {B^2} {4 A^2} + \frac {C^2} {4 A^2} - \frac D A$

$\ds $

$=$

$\ds \frac {B^2} {4 A^2} + \frac {C^2} {4 A^2} - \frac {4 A D} {4 A^2}$

$\ds $

$=$

$\ds \frac 1 {4 A^2} \paren {B^2 + C^2 - 4 A D}$

This last expression is non-negative if and only if $B^2 + C^2 \ge 4 A D$.

In such a case $\dfrac 1 {4 A^2} \paren {B^2 + C^2 - 4 A D}$ is in the form $R^2$ and so:

$\paren {x + \dfrac B {2 A} }^2 + \paren {y + \dfrac C {2 A} }^2 = \dfrac 1 {4 A^2} \paren {B^2 + C^2 - 4 A D}$

is in the form:

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

$\blacksquare$