Circles SAT question

Choice D is correct. The graph in the xy-plane of an equation of the form ${(x-h)}^2+{(y-k)}^2=r^2$ is a circle with center $(h,k)$ and a radius of length $r$. It's given that circle A is represented by $x^2+{(y-1)}^2=49$, which can be rewritten as $x^2+{(y-1)}^2=7^2$. Therefore, circle A has center $(0,1)$ and a radius of length $7$. Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point $(x,y)$ on circle A, the point $(x,y-2)$ lies on circle B. Moreover, if $(h,k)$ is the center of circle A, then $(h,k-2)$ is the center of circle B. Therefore, circle B has a radius of $7$ and the center of circle B is $(0,1-2)$, or $(0,-1)$. Thus, circle B can be represented by the equation $x^2+{(y+1)}^2=7^2$, or $x^2+{(y+1)}^2=49$.

Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right $2$ units.

Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up $2$ units.

Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left $2$ units.