SAT Question 6585d841: Circles answer and explanation with Desmos solution

Choice C is correct. It's given that in the xy-plane, the graph of the given equation is a circle. The equation of a circle in the xy-plane can be written in the form ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where $\left(h,k\right)$ is the center of the circle and $r$ is the length of the circle's radius. Subtracting $6y$ from both sides of the equation $x^2+14x+y^2=6y+109$ yields $x^2+14x+y^2-6y=109$. By completing the square, this equation can be rewritten as $\left(x^2+14x+{\left(\frac{14}{2}\right)}^2\right)+\left(y^2-6y+{\left(\frac{-6}{2}\right)}^2\right)=109+{\left(\frac{14}{2}\right)}^2+{\left(\frac{-6}{2}\right)}^2$. This equation can be rewritten as $\left(x^2+14x+49\right)+\left(y^2-6y+9\right)=109+49+9$, or ${\left(x+7\right)}^2+{\left(y-3\right)}^2=167$. Therefore, $r^2=167$. Taking the square root of both sides of this equation yields $r=\sqrt{167}$ and $r=-\sqrt{167}$. Since $r$ is the length of the circle's radius, $r$ must be positive. Therefore, the length of the circle's radius is $\sqrt{167}$.