Gleaning Info from the General Equation of a Circle
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In our lesson on this topic, we made an observation about the relationship between the coordinates of the center of the circle and the values of C and D in the general equation form of a circle.  Take another look ...

2.  How do the coordinates of the center of a circle relate to C and D when the equation of the circle is in the general form
                                      ?

Let's make some observations.  Re-examine our previous equations in general form and center-radius form.  Do you see a relationship between the center coordinates and C and D?

General form Center-radius form

C = -4,  D = 10
Center (2, -5)

C = -4,  D = -6
Center (2, 3)

It appears that the values of C and D are (-2) times the coordinates of the center respectively.  Why is this occurring?
When is expanded, becomes , where the center term's coefficient doubles the value of -2.  Remember that while the equation deals with , the actual x-coordinate of the center of this circle is +2.


 

Now, we could rewrite these findings in this form, letting C = 2p and D = 2q:


where the center of the circle is (-p, -q).

 

Can we also find a way, using this rewrite, to determine the radius of the circle directly from the general form?

Let's complete the square on this equation and see what happens to the radius.

Center:  (-p, -q)

Radius:
 

We have now found an expression to represent the radius, based upon our rewrite of the general form equation:
Radius: