SAT Question 2266984b: Circles answer and explanation with Desmos solution

Choice B is correct. The standard equation of a circle in the $xy$-plane is of the form $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ are the coordinates of the center of the circle and $r$ is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, $x^2+20x$, needs a $100$ to complete the square, and the sum of the second two terms, $y^2+16y$, needs a $64$ to complete the square. Adding $100$ and $64$ to both sides of the given equation yields $(x^2+20x+100)+(y^2+16y+64)=-20+100+64$, which is equivalent to $(x+10{)}^2+(y+8{)}^2=144$. Therefore, the coordinates of the center of the circle are $(-10,-8)$.