SAT Question fa2771d5: Circles answer and explanation with Desmos solution

Choice C is correct. The equation of a circle in the xy-plane can be written as $(x-h{)}^2+(y-k{)}^2=r^2$, where the center of the circle is $(h,k)$ and the radius of the circle is $r$ units. It’s given that circle A has the equation $(x-7{)}^2+(y+3{)}^2=1$, which can be written as $(x-7{)}^2+(y-(-3){)}^2=1^2$. It follows that $h=7$, $k=-3$, and $r=1$. Therefore, the center of circle A is $(7,-3)$ and its radius is $1$ unit. If circle A is translated $4$ units to the right, the x-coordinate of the center will increase by $4$, while the y-coordinate and the radius of the circle will remain unchanged. Translating the center of circle A to the right $4$ units yields $(7+4,-3)$, or $(11,-3)$. Therefore, the center of circle B is $(11,-3)$. Substituting $11$ for $h$, $-3$ for $k$, and $1$ for $r$ into the equation $(x-h{)}^2+(y-k{)}^2=r^2$ yields $(x-11{)}^2+(y-(-3){)}^2=1$, or $(x-11{)}^2+(y+3{)}^2=1$. Therefore, the equation $(x-11{)}^2+(y+3{)}^2=1$ represents circle B.