Let's make some observations about the general form of a circle.
There are some interesting connections that pop up in this form.
General Form of a Circle:
x^{2} + y^{2} + Cx + Dy + E = 0 

Is there a relationship between the coordinates of the center of a circle and the values of C and D in the general form of a circle? Take a look!
General Form 
CenterRadius Form 
x^{2} + y^{2} + 2x  4y  11 = 0
C = 2, D = 4

(x + 1)^{2} + (y  2)^{2} = 16
Center (1,2) 
x^{2} + y^{2}  4x  6y + 8 = 0
C = 4, D = 6 
(x  2)^{2} + (y  3)^{2} = 5
Center (2,3) 


It appears that the values of C and D are (2) times the coordinates of the center respectively.
Is this really occurring?
When (x + 1)^{2} + (y  2)^{2} = 16 is expanded, (x + 1)^{2} becomes x^{2} + 2x + 1,
where the center
term's coefficient doubles the value of +1.
While the equation deals
with (x + 1)^{2}, the actual xcoordinate of the center of this circle is (1).
Yes, this relationship is occurring!
Now, let's see if this relationship will allow us to establish any other relationships regarding the general form of a circle and the centerradius form.
Let the center of the circle be (p, q).
We now know C = (2)(p) = 2p and D = (2)(q) = 2q, so we have:
x^{2} + y^{2} + 2px + 2qy + E = 0
If we complete the square on this equation, we will make a discovery about the radius.

Center: (p,q)
Radius: 
We have now found a relationship between the general form of the equation and radius.
Radius:
Remember, you have already seen how to find the center of the circle from the general form of the equation.
So, now we have a way to obtain the center and the radius directly from the general form of the equation of a circle.
Is this easier than completing the square to get the centerradius form of the equation to get the center and radius? Probably not, since you would have to remember these relationships.
But, isn't it amazing what a little investigating and a little algebra can uncover!
