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

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center 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-15"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

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

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 $100$ to complete the square, and the sum of the second two terms, $y^{2}+16y$, needs $64$ 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 $(-10,-8)$

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete thesquares or when identifying the coordinates of the center.

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

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

","suggestedAnswer":[{"@type":"Answer","position":0,"encodingFormat":"text/html","text":"

$(-20,-16)$

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

$(10,8)$

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

$(20,16)$

"}],"acceptedAnswer":{"@type":"Answer","position":1,"encodingFormat":"text/html","text":"

$(-10,-8)$

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

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete thesquares or when identifying the coordinates of the center.

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

Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete thesquares or when identifying the coordinates of the center.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-15"}

$x^{2}+20x+y^{2}+16y=-20$The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

$x^{2}+20x+y^{2}+16y=-20$The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

$x^{2}+20x+y^{2}+16y=-20$The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?

$x^{2}+20x+y^{2}+16y=-20$

The equation above defines a circle in the $xy$-plane. What are the coordinates of the center of the circle?