{"@context":"https://schema.org/","@type":"Quiz","name":"

In the $xy$-plane, the graph of $2x^{2}-6x+2y^{2}+2y=45$ is a circle. What is the radius of the circle?

","hasPart":{"@type":"Question","typicalAgeRange":"12-22","educationalAlignment":[{"@type":"AlignmentObject","alignmentType":"educationalSubject","targetName":"SAT"},{"@type":"AlignmentObject","alignmentType":"educationalLevel","targetName":"SAT","educationalFramework":"Common Core","targetUrl":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-27"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

In the $xy$-plane, the graph of $2x^{2}-6x+2y^{2}+2y=45$ is a circle. What is the radius of the circle?

","comment":{"@type":"Comment","text":"

Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard form, $(x-h)^{2}+(y-k)^{2}=t^{2}$, 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 $5$.

Choices B, C, and D are incorrect and may result from errors in manipulating the equation or a misconception about the standard form of the equation of a circle in the $xy$-plane.

"},"encodingFormat":"text/markdown","text":"

In the $xy$-plane, the graph of $2x^{2}-6x+2y^{2}+2y=45$ is a circle. What is the radius of the circle?

","suggestedAnswer":[{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$6.5$

"},{"@type":"Answer","position":2,"encodingFormat":"text/html","text":"

$\\sqrt{40}$

"},{"@type":"Answer","position":3,"encodingFormat":"text/html","text":"

$\\sqrt{50}$

"}],"acceptedAnswer":{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$5$

","comment":{"@type":"Comment","text":"

Choices B, C, and D are incorrect and may result from errors in manipulating the equation or a misconception about the standard form of the equation of a circle in the $xy$-plane.

"},"answerExplanation":{"@type":"Comment","text":"

Choices B, C, and D are incorrect and may result from errors in manipulating the equation or a misconception about the standard form of the equation of a circle in the $xy$-plane.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-27"}