Consider two points C(h,k) and P(x,y) ;and the distance between them is r. From the distance formula:

square both sides
r²= ( x - h )²+ ( y - k )²

r²= ( x - h )² + ( y - k )² is the standard equation for circles.

Ax²+ Ay²+ Dx + Ey + F = 0
is the general form of the equation for circles.

ex.
Write the general equation of the circle with center at C(4,-5) and a
radius of 5.

r²= ( x - h )²+ ( y - k )²
5²= ( x - 4 )²+ ( y -(-5))²
25 = x² -8x + 16 + y² + 10y + 25

x²+ y² -8x + 10y + 16 = 0 is the general equation.

Find the center and the radius of the circle with equation x²+ y²-10x - 4y + 16 = 0.

x²+ y²-10x - 4y +16 = 0

(x-10x ) + (y- 4y ) = -16 complete the square.
(x²-10x + 25) + (y²- 4y + 4) = -16 + 25 + 4
( x -5)² + ( y - 2 )² = 13

The center is at (5, 2) and the radius is

If the constant term in the standard equation is positive, then a graph of a circle exist. If it is zero, a single point exist, If it is negative, there is no graph.