Circles SAT question

Choice A is correct. The standard form of an equation of a circle in the xy-plane is ${(x-h)}^2+{(y-k)}^2=r^2$, where the coordinates of the center of the circle are $(h,k)$ and the length of the radius of the circle is $r$. The equation of circle A, $x^2+{(y-2)}^2=36$, can be rewritten as ${(x-0)}^2+{(y-2)}^2=6^2$. Therefore, the center of circle A is at $(0,2)$ and the length of the radius of circle A is $6$. If circle A is shifted down $4$ units, the y-coordinate of its center will decrease by $4$; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at $(0,2-4)$, or $(0,-2)$, and its radius is $6$. Substituting $0$ for $h$, $-2$ for $k$, and $6$ for $r$ in the equation ${(x-h)}^2+{(y-k)}^2=r^2$ yields ${(x-0)}^2+{(y-(-2))}^2={(6)}^2$, or $x^2+{(y+2)}^2=36$. Therefore, the equation $x^2+{(y+2)}^2=36$ represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, $4$ units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, $4$ units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, $4$ units.