SAT Mathematics Practice Test 6

One way to find the radius of the circle is to put the given equation in standard form, ${\left(\mathit{x}-\mathit{h}\right)}^2+{\left(\mathit{y}-\mathit{k}\right)}^2={\mathit{r}}^2$,where $\left(\mathit{h},\mathit{k}\right)$ is the center of the circle and the radius of the circle is r. To do this, divide the original equation,$2{\mathit{x}}^2-6\mathit{x}+2{\mathit{y}}^2+2\mathit{y}=45$, by 2 to make the leading coefficients of ${\mathit{x}}^2$ and ${\mathit{y}}^2$ each equal to $1:{\mathit{x}}^2-3\mathit{x}+{\mathit{y}}^2+\mathit{y}=22.5$. Then complete the square to put the equation in standard form. To do so, first rewrite ${\mathit{x}}^2-3\mathit{x}+{\mathit{y}}^2+\mathit{y}=22.5$ as .Second, add $2.25$ and $0.25$ to both sides of the equation: $\left({\mathit{x}}^2-3\mathit{x}+2.25\right)+\left({\mathit{y}}^2+\mathit{y}+0.25\right)=25$. Since and $25=5^2$,it follows that ${\left(\mathit{x}-1.5\right)}^2+{\left(\mathit{y}-0.5\right)}^2=5^2$Therefore, the radius of the circle is 5.
Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the xy-plane.