SAT Question 6b4707aa: Circles answer and explanation with Desmos solution

An equation of a circle in the xy-plane can be written as $(x-t{)}^2+(y-u{)}^2=r^2$, where the center of the circle is $(t,u)$, 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 $(x+5{)}^2+(y-5{)}^2=25$, which is equivalent to $(x+5{)}^2+(y-5{)}^2=5^2$. Therefore, the center of circle A is $(-5,5)$ and the radius of circle A is $5$. 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 $(-5,5)$ and the radius of circle B is $2(5)$, or $10$. Substituting $-5$ for $t$, $5$ for $u$, and $10$ for $r$ in the equation $(x-t{)}^2+(y-u{)}^2=r^2$ yields $(x+5{)}^2+(y-5{)}^2=10^2$, which is equivalent to $(x+5{)}^2+(y-5{)}^2=100$. It follows that the equation of circle B in the xy-plane is $(x+5{)}^2+(y-5{)}^2=100$. It's also given that the equation defining circle B in the xy-plane is $(x+5{)}^2+(y-5{)}^2=k$, where $k$ is a constant. Therefore, the value of $k$ is $100$.