SAT Question 9e44284b: Circles answer and explanation with Desmos solution

Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, $(x-h{)}^2+(y-k{)}^2=r^2$, where $(h,k)$ is the center of the circle and the radius of the circle is $r$. To do this, divide the original equation, $2x^2-6x+2y^2+2y=45$, by $2$ to make the leading coefficients of $x^2$ and $y^2$ each equal to $1$: $x^2-3x+y^2+y=22.5$. Then complete the square to put the equation in standard form. To do so, first rewrite $x^2-3x+y^2+y=22.5$ as $(x^2-3x+2.25)-2.25+(y^2+y+0.25)-0.25=22.5$. Second, add $2.25$ and $0.25$ to both sides of the equation: $(x^2-3x+2.25)+(y^2+y+0.25)=25$. Since $x^2-3x+2.25=(x-1.5{)}^2$, $y^2+y+0.25=(y+0.5{)}^2$, and $25=5^2$, it follows that $(x-1.5{)}^2+(y+0.5{)}^2=5^2$. Therefore, the radius of the circle is $5$.