The general equation of a circle with center (h, k) and radius r is (x - h)^2 + (y - k)^2 = r^2.
The circle with equation x^2 + y^2 = 18 is moved so that the center lies at (2,3). The equation of the circle is now (x - 2)^2 + (y - 3)^2 = 18
=> x^2 - 4x + 4 + y^2 - 6y + 9 = 18
=> x^2 + y^2 - 4x - 6y = 5
The graph of the original circle (in black) and and that of the shifted circle (in red) is given below:
The equation of the circle after it has been shifted is: x^2 + y^2 - 4x - 6y = 5