sat-dog

The correct answer is $16$. An equation of a circle in the xy-plane can be written as ${\left(x-t\right)}^2+{\left(y-u\right)}^2=r^2$, where the center of the circle is $\left(t,u\right)$ , the radius of the circle is $r$, and where $t$, $u$, and $r$ are constants. It’s given that the equation of circle A is ${\left(x+5\right)}^2+{\left(y-5\right)}^2=4$, which is equivalent to ${\left(x+5\right)}^2+{\left(y-5\right)}^2=2^2$. Therefore, the center of circle A is $\left(-5,5\right)$ and the radius of circle A is $2$. It’s given that circle B has the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is $\left(-5,5\right)$ and the radius of circle B is $2\left(2\right)$, or $4$. Substituting $-5$ for $t$, $5$ for $u$, and $4$ for $r$ into the equation ${\left(x-t\right)}^2+{\left(y-u\right)}^2=r^2$  yields ${\left(x+5\right)}^2+{\left(y-5\right)}^2=4^2$, which is equivalent to ${\left(x+5\right)}^2+{\left(y-5\right)}^2=16$. It follows that the equation of circle B in the xy-plane is ${\left(x+5\right)}^2+{\left(y-5\right)}^2=16$. Therefore, the value of $k$ is $16$.