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

square both sides

r^{2}= ( x - h )^{2}+ ( y - k )^{2}

r^{2}= ( x - h )^{2 }+ ( y - k )^{2} is the standard equation for circles.

Ax^{2}+ Ay^{2}+ 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.

r^{2}= ( x - h )^{2}+ ( y - k )^{2}

5^{2}= ( x - 4 )^{2}+ ( y -(-5))^{2}

25 = x^{2} -8x + 16 + y^{2} + 10y + 25

x^{2}+ y^{2 }-8x + 10y + 16 = 0 is the general equation.

Find the center and the radius of the circle with equation x^{2}+ y^{2}-10x - 4y + 16 = 0.

x^{2}+ y^{2}-10x - 4y +16 = 0

(x-10x ) + (y- 4y ) = -16 complete the square.

(x^{2}-10x + 25) + (y^{2}- 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.