circumference

The circumference is the geometric place of all points in a plane that are equidistant from a fixed point called centre. The fixed distance between the points and the centre is called radius.
If C(c₁,c₂) and the radius is r:

Example 1: if C(1,3) and r = 3, then the equation is:
(x – 1)²+ (y – 3)² = 3² → x² + y² – 2x – 6y + 1 = 0

Example 2: find the center and the radius of the circumference: x² + y² + 2x – 8y - 8 = 0
We use the method of “completing squares”, by using the remarkable identities:
x² + 2x + 1 + y² – 8y + 16 - 25 = 0
(x + 1)²+(y – 4)² = 5² → C(-1,4)    r = 5

Exercises:

1.- Calculate the equation of the circumference with center (1,-2) and radius 4.

2.- Calculate the coordinates of the center and the radius of the circumference x² + y² + 6x - 10y - 66 = 0

Solutions: 1) (x - 1)² + (y + 2)² = 4²; 2) C(-3,5) r = 10