1.3.3—Circle equation

The circle with centre (a,b) and radius r has (x−a)²+(y−b)²=r². Expanding gives a general quadratic form, but completing squares recovers the geometry.

Use the radius as a distance, check r²>0, and substitute a point to test whether it lies on the circle. A zero radius is a single point, not a proper circle.

x²+y²−6x+4y−3=0 becomes (x−3)²+(y+2)²=16, so centre (3,−2) and radius 4.

The signs in the centre reverse when completing squares: (x−3)² gives centre x=3, while (y+2)² gives y=−2.