Let the center of the circle be (h,k). 
   

The red line is the given line
5x-2y-1 = 0.  The green line is the line perpendicular to it.

Since 5x-2y-1 = 0 solved for y is , its slope
is  so the green line has slope  and since it
goes through (1,2) we use the point slope formula and get the equation:






 <-- equation of green line.

Since (h,k) lies on the green line,



Next we use the formula for the distance from a point (x1,y1) to the line
Ax+By+C=0, which is



and the distance from (h,k) must be ±3 units to the line 5x-2y-1=0, so we have









We solve the system:



using the + and then the -, the two centers are

(h,k) = (-1.785439973,3.114172029)

and

(h,k) = (3.785430073,0.8858279709)

Edwin