your equation of circle is
that will be one with center in (0,0)
but equation of sifted circle is
so to see where your center is just :
so center is in (a,b)
both formulas are the same, just in first one ( with center circle) you have that your a=0 and b=0 meaning that it's not shifted ...
P.S. your center up there is not as you write (-2,-3) it's (2,-3) but that just typo i think