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

The equation $x^{2}+\\left(y-2\\right)^{2}=36$ represents circle A. Circle B is obtained by shifting circle A down 4 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-44"}],"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 $x^{2}+\\left(y-2\\right)^{2}=36$ represents circle A. Circle B is obtained by shifting circle A down 4 units in the $xy$-plane. Which of the following equations represents circle B?

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

Choice A is correct. The standard form of an equation of a circle in the $xy$-plane is $\\left(\\bm{x}-\\bm{h}\\right)^{2}+\\left(\\bm{y}-\\bm{k}\\right)^{2}=\\bm{r}^{2}$, where the coordinates of the center of the circle are $\\left(\\bm{h},\\bm{k}\\right)$ and the length of the radius of the circle is $\\bm{r}$. The equation of circle A, $\\bm{x}^{2}+\\left(\\bm{y}-\\bm{2}\\right)^{2}=36$, can be rewritten as $\\left(\\bm{x}-\\bm{0}\\right)^{2}+\\left(\\bm{y}-\\bm{2}\\right)^{2}=6^{2}$. Therefore, the center of circle A is at $\\left(0,2\\right)$ and the length of the radius of circle A is $6$. If circle A is shifted down $4$ units, the $y$-coordinate of its center will decrease by $4$; the radius of the circle and the $x$-coordinate of its center will not change. Therefore, the center of circle B is at $\\left(0,2-4\\right)$, or $\\left(0,-2\\right)$, and its radius is $6$. Substituting $0$ for $h$, $-2$ for $\\bm{k}$, and $6$ for $\\bm{r}$ in the equation $\\left(\\bm{x}-\\bm{h}\\right)^{2}+\\left(\\bm{y}-\\bm{k}\\right)^{2}=\\bm{r}^{2}$ yields $\\left(\\bm{x}-\\bm{0}\\right)^{2}+\\left(\\bm{y}-\\left(-2\\right)\\right)^{2}=\\left( 6\\right)^{2}$, or $\\bm{x}^{2}+\\left(\\bm{y}+\\bm{2}\\right)^{2}=36$. Therefore, the equation $\\bm{x}^{2}+\\left(\\bm{y}+\\bm{2}\\right)^{2}=36$ represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.

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

The equation $x^{2}+\\left(y-2\\right)^{2}=36$ represents circle A. Circle B is obtained by shifting circle A down 4 units in the $xy$-plane. Which of the following equations represents circle B?

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

$x^{2}+(y-6)^{2}=36$

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

$(x-4)^{2}+(y-2)^{2}=36$

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

$(x+4)^{2}+(y-2)^{2}=36$

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

$x^{2}+(y+2)^{2}=36$

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

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.

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

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, 4 units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, 4 units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, 4 units.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-44"}

The equation $x^{2}+\left(y-2\right)^{2}=36$ represents circle A. Circle B is obtained by shifting circle A down 4 units in the $xy$-plane. Which of the following equations represents circle B?