Circle forms and real-circle conditions

Dividing by the common non-zero squared-term coefficient is essential before completing squares. A circle is the set of points at one positive distance from its centre; the distance formula produces the sum of two squares. x2+y2+2gx+2fy+c=0 becomes (x+g)2+(y+f)2=g2+f2-c. Read centre (-g,-f) and radius sqrt(g2+f2-c). Method: normalise; complete each square; identify the centre; check the sign of the squared radius. A positive right side gives a real circle, zero gives one point, and a negative right side gives no real points. Example: 2x2+2y2-8x+4y=0 becomes (x-2)2+(y+1)2=5. The centre is (2,-1) and radius 5. Exam check: the constant on the right is the squared radius, not the radius. If it equals zero, do not report a circle of positive radius.