Equation of Circle/Cartesian/Formulation 3

Equation of Circle in Cartesian Plane: Also presented as

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:

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

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

$=$

$\ds R^2$

$\ds \leadsto \ \ $

$\ds x^2 - 2 a x + a^2 + y^2 - 2 b y + b^2$

$=$

$\ds R^2$

multiplying out

$\ds \leadsto \ \ $

$\ds x^2 + y^2 - 2 a x - 2 b y + a^2 + b^2 - R^2$

$=$

$\ds 0$

rearranging

$\ds \leadsto \ \ $

$\ds x^2 + y^2 - 2 a x - 2 b y + c$

$=$

$\ds 0$

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

$\blacksquare$

1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $14$. To find the equation of the circle whose centre is $\tuple {\alpha, \beta}$ and radius $r$