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

Which of the following equations represents a circle in the $xy$-plane that intersects the $y$-axis at exactly one point?

","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-41"}],"about":{"@type":"Thing","name":"Math, Geometry and Trigonometry, Duplicate"},"educationalLevel":"intermediate","eduQuestionType":"Multiple choice","learningResourceType":"Practice problem","assesses":["Math","Geometry and Trigonometry","Duplicate"],"name":"

Which of the following equations represents a circle in the $xy$-plane that intersects the $y$-axis at exactly one point?

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

Choice C is correct. The graph of the equation $\\left(x-h\\right)^{2}+\\left(y-k\\right)^{2}=r^{2}$ in the $xy$-plane is a circle with center $(h,k)$ and a radius of length $r$. The radius of a circle is the distance from the center of the circle to any point on the circle. If a circle in the $xy$-plane intersects the $y$-axis at exactly one point, then the perpendicular distance from the center of the circle to this point on the $y$-axis must be equal to the length of the circle's radius. It follows that the $x$-coordinate of the circle's center must be equivalent to the length of the circle's radius. In other words, if the graph of $(x-h)^{2}+\\left(y-k\\right)^{2}=r^{2}$ is a circle that intersects the $y$-axis at exactly one point, then $r=\\left|h\\right|$ must be true. The equation in choice C is $(x-4)^{2}+\\left(y-9\\right)^{2}=16$, or $\\left(x-4\\right)^{2}+\\left(y-9\\right)^{2}=4^{2}$. This equation is in the form $\\left(x-h\\right)^{2}+\\left(y-k\\right)^{2}=r^{2}$, where $h=4$, $k=9$, and $r=4$, and represents a circle in the $xy$-plane with center $(4,9)$ and radius of length $4$. Substituting $4$ for $r$ and $4$ for $h$ in the equation $r=|h|$ yields $4=|4|$, or $4=4$, which is true.Therefore, the equation in choice C represents a circle in the $xy$-plane that intersects the $y$-axis at exactly one point.

Choice A is incorrect. This is the equation of a circle that does not intersect the $y$-axis at any point.

Choice B is incorrect. This is an equation of a circle that intersects the $x$-axis, not the $y$-axis, at exactly one point.

Choice D is incorrect. This is the equation of a circle with the center located on the $y$-axis and thus intersects the $y$-axis at exactly two points, not exactly one point.

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

Which of the following equations represents a circle in the $xy$-plane that intersects the $y$-axis at exactly one point?

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

$(x-8)^{2}+(y-8)^{2}=16$

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

$(x-8)^{2}+(y-4)^{2}=16$

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

$x^{2}+(y-9)^{2}=16$

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

$(x-4)^{2}+(y-9)^{2}=16$

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

Choice A is incorrect. This is the equation of a circle that does not intersect the $y$-axis at any point.

Choice B is incorrect. This is an equation of a circle that intersects the $x$-axis, not the $y$-axis, at exactly one point.

Choice D is incorrect. This is the equation of a circle with the center located on the $y$-axis and thus intersects the $y$-axis at exactly two points, not exactly one point.

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

Choice A is incorrect. This is the equation of a circle that does not intersect the $y$-axis at any point.

Choice B is incorrect. This is an equation of a circle that intersects the $x$-axis, not the $y$-axis, at exactly one point.

Choice D is incorrect. This is the equation of a circle with the center located on the $y$-axis and thus intersects the $y$-axis at exactly two points, not exactly one point.

"}}},"url":"https://classnotes.xyz/mcqs/sat-math-geometry-and-trigonometry-multiple-choice-question-41"}