Coordinates

Add 1 to complete the square on x: $x^2-2x +1 + y^2 = 25$

Complete the square: $(x-1)^2 + y^2 = 25$

The center occurs at $\color{brown}\boxed{(1,0)}$, (x-y pair when the value of the circle equals 0). 

Also, note that the constant on the right-hand side of the equation equals the radius squared, so the radius is $\sqrt {25} = \color{brown}\boxed 5$