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The equation $x^{2}+\\left(y-1\\right)^{2}=49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the $xy$-plane. Which of the following equations represents circle B?

","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-37"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

The equation $x^{2}+\\left(y-1\\right)^{2}=49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the $xy$-plane. Which of the following equations represents circle B?

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

Choice D is correct. The graph in the $xy$-plane of an equation of the form $\\left(\\bm{x}-\\bm{h}\\right)^{2}+\\left(y-\\bm{k}\\right)^{2}=r^{2}$ is a circle with center $\\left(\\bm{h},\\bm{k}\\right)$ and a radius of length $r$. It's given that circle A is represented by $\\bm{x}^{2}+\\left(y-1\\right)^{2}=49$, which can be rewritten as $\\bm{x}^{2}+\\left(\\bm{y}-1\\right)^{2}=7^{2}$. Therefore, circle A has center $\\left(0,1\\right)$ and a radius of length $7$. Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point $\\left(\\bm{x},\\bm{y}\\right)$ on circle A, the point $\\left(\\bm{x},\\bm{y}-2\\right)$ lies on circle B. Moreover, if $\\left(\\bm{h},\\bm{k}\\right)$ is the center of circle A, then $\\left(\\bm{h},\\bm{k}-2\\right)$ is the center of circle B. Therefore, circle B has a radius of $7$ and the center of circle B is $\\left(0,1-2\\right)$, or $\\left(0,-1\\right)$. Thus, circle B can be represented by the equation $\\bm{x}^{2}+\\left(\\bm{y}+1\\right)^{2}=7^{2}$, or $\\bm{x}^{2}+\\left(\\bm{y}+1\\right)^{2}=49$.

Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right $2$ units.

Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up $2$ units.

Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left $2$ units.

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The equation $x^{2}+\\left(y-1\\right)^{2}=49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the $xy$-plane. Which of the following equations represents circle B?

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

$(x-2)^{2}+(y-1)^{2}=49$

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

$x^{2}+(y-3)^{2}=49$

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

$(x+2)^{2}+(y-1)^{2}=49$

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

$x^{2}+(y+1)^{2}=49$

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

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"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-37"}

The equation $x^{2}+\left(y-1\right)^{2}=49$ represents circle A. Circle B is obtained by shifting circle A down $2$ units in the $xy$-plane. Which of the following equations represents circle B?