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The Cartesian Circle

Written by tutor Steve C.

There are three locations for graphing a circle in the XY Cartesian Plane:

At the Origin, On the Edge, and Anyplace Else.

Here is the standard circle with center at the origin, defined by x2 + y2 = 16

Standard Circle

The general form is actually x2 + y2 = r2 where the radius r = 4

Here is the same size circle with center at (5, 5), defined by (x-5)2 + (y-5)2 = 16

The general form is actually

(x-a)2 + (y-b)2 = r2

where the center is (a, b).


Notice that the center points here are positive. The standard form notation of the circle equation does allow for them to be shown as negative.
Standard Circle at 5,5
If the circle center is at (-5, -5)

then the standard form of the circle

becomes (x+5)2 + (y+5)2 = 16


A similar pattern will result if the circle is moved to the 2nd quadrant where a<0 and="" b="">0, or if it is moved to the 4th quadrant where a>0 and b<0.>
Standard Circle at -5,-5

The Special Case

The final location for a circle graph is where the edge falls along the x axis and y axis.

Here is the same size circle with center at (4, 4), defined by (x-4)2 + (y-4)2 = 16.

The standard form of the equation is still (x-a)2 + (y-b)2 = r2 .

However, in this case, a = b = r.

In fact, we can state that the graph of the equation in this form

(x-r)2 + (y-r)2 = r2

Is a circle sitting on the edge of the x axis and the y axis.
Standard Circle at 4,4
We can carry this further,
(for the 1st quadrant only)
where (x-a)2 + (y-b)2 = r2 :

if a=r, the circle sits on the x axis;
if a > r, the circle sits away from the x axis;
if a < r,="" the="" circle="" crosses="" the="" x="" axis="">

if b=r, the circle sits on the y axis;
if b > r, the circle sits away from the y axis;
if b < r,="" the="" circle="" crosses="" the="" y="" axis="" twice.="">
Comparison in Special Case of Circles
         (x-2)2 + (y-4)2 = 42                              (x-4)2 + (y-2)2 = 42