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

A circle in the $xy$-plane has its center at $(-4,5)$ and the point $(-8,8)$ lies on the circle. Which equation represents this 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-60"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

A circle in the $xy$-plane has its center at $(-4,5)$ and the point $(-8,8)$ lies on the circle. Which equation represents this circle?

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

Choice D is correct. A circle in the $xy$-plane can be represented by an equation of 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 a radius of the circle. It's given that the circle has its center at $\\left(-4,5\\right)$. Therefore, $h=-4$ and $k=5$. Substituting $-4$ for $h$ and $5$ for $k$ in the equation $\\left(x-h\\right)^{2}+\\left(y-k\\right)^{2}=r^{2}$ yields $\\left(x-\\left(-4\\right)\\right)^{2}+\\left(y-5\\right)^{2}=r^{2}$, or $\\left(x+4\\right)^{2}+\\left(y-5\\right)^{2}=r^{2}$. It's also given that the point $\\left(-8,8\\right)$ lies on the circle. Substituting $-8$ for $x$ and $8$ for $y$ in the equation $\\left(x+4\\right)^{2}+\\left(y-5\\right)^{2}=r^{2}$ yields $\\left(-8+4\\right)^{2}+\\left(8-5\\right)^{2}=r^{2}$, or $\\left(-4\\right)^{2}+\\left(3\\right)^{2}=r^{2}$, which is equivalent to $16+9=r^{2}$, or $25=r^{2}$. Substituting $25$ for $r^{2}$ in the equation $(x+4)^{2}+(y-5)^{2}=r^{2}$ yields $(x+4)^{2}+(y-5)^{2}=25$. Thus, the equation $(x+4)^{2}+(y-5)^{2}=r^{2}$ represents the circle.

Choice A is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

Choice B is incorrect. The point $(-8,8)$ doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

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

A circle in the $xy$-plane has its center at $(-4,5)$ and the point $(-8,8)$ lies on the circle. Which equation represents this circle?

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

$(x-4)^{2}+(y+5)^{2}=5$

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

$(x+4)^{2}+(y-5)^{2}=5$

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

$(x-4)^{2}+(y+5)^{2}=25$

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

$(x+4)^{2}+(y-5)^{2}=25$

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

Choice A is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

Choice B is incorrect. The point $(-8,8)$ doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

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

Choice A is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

Choice B is incorrect. The point $(-8,8)$ doesn't lie on the circle represented by this equation.

Choice C is incorrect. The circle represented by this equation has its center at $(4,-5)$, not $(-4,5)$, and the point $(-8,8)$ doesn't lie on the circle.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-60"}