SAT Question a38c0183: Circles answer and explanation with Desmos solution

Choice A is correct. According to the graph, the center of circle A has coordinates $\left(-2,0\right)$, and the radius of circle A is $3$. It's given that circle B is the result of shifting circle A down $6$ units and increasing the radius so that the radius of circle B is $2$ times the radius of circle A. It follows that the center of circle B is $6$ units below the center of circle A. The point that's $6$ units below $\left(-2,0\right)$ has the same x-coordinate as $\left(-2,0\right)$ and has a y-coordinate that is $6$ less than the y-coordinate of $\left(-2,0\right)$. Therefore, the coordinates of the center of circle B are $\left(-2,0-6\right)$, or $\left(-2,-6\right)$. Since the radius of circle B is $2$ times the radius of circle A, the radius of circle B is $\left(2\right)\left(3\right)$. A circle in the xy-plane can be defined by an equation of the form ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where the coordinates of the center of the circle are $\left(h,k\right)$ and the radius of the circle is $r$. Substituting $-2$ for $h$, $-6$ for $k$, and $\left(2\right)\left(3\right)$ for $r$ in this equation yields ${\left(x-\left(-2\right)\right)}^2+{\left(y-\left(-6\right)\right)}^2={\left(\left(2\right)\left(3\right)\right)}^2$, which is equivalent to ${\left(x+2\right)}^2+{\left(y+6\right)}^2={\left(2\right)}^2{\left(3\right)}^2$, or ${\left(x+2\right)}^2+{\left(y+6\right)}^2=\left(4\right)\left(9\right)$. Therefore, the equation ${\left(x+2\right)}^2+{\left(y+6\right)}^2=\left(4\right)\left(9\right)$ defines circle B.