{"id":"cb_question_bank:a904ec9c-4c1f-447f-91ee-8fef470bedbb","source_id":"cb_question_bank","source_item_id":"a904ec9c-4c1f-447f-91ee-8fef470bedbb","external_id":"78306704-22d8-4501-a060-e7b7995c3ceb","ibn":null,"question_id":"9d159400","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Which of the following equations represents a circle in the xy -plane that intersects the y -axis at exactly one point?","stimulus":"","stimulus_html":null,"stem_html":"

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

","rationale_html":"

Choice C is correct. The graph of the equation x-h2+y-k2=r2 in the xy-plane is a circle with center h,k and a radius of length r\n. 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-h2+y-k2=r2 is a circle that intersects the y-axis at exactly one point, then r=h must be true. The equation in choice C is x-42+y-92=16, or x-42+y-92=42. This equation is in the form x-h2+y-k2=r2, 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\n. Substituting 4\n for r\n and 4\n for h\n 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. 

\n

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

\n

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

\n

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.

","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.840000+00:00","source_updated_at":"2023-08-02T20:25:59.840000+00:00","options":[{"label":"A","content_html":"

\n\t\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\tx\n\t\t\t\t\t-\n\t\t\t\t\t8\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\ty\n\t\t\t\t\t-\n\t\t\t\t\t8\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\n\t=\n\t16\n\n

","ord":0},{"label":"B","content_html":"

\n\t\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\tx\n\t\t\t\t\t-\n\t\t\t\t\t8\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\ty\n\t\t\t\t\t-\n\t\t\t\t\t4\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\n\t=\n\t16\n\n

","ord":1},{"label":"C","content_html":"

\n\t\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\tx\n\t\t\t\t\t-\n\t\t\t\t\t4\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\ty\n\t\t\t\t\t-\n\t\t\t\t\t9\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\n\t=\n\t16\n\n

","ord":2},{"label":"D","content_html":"

\n\t\n\t\t\n\t\t\tx\n\t\t\t2\n\t\t\n\t\t+\n\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\ty\n\t\t\t\t\t-\n\t\t\t\t\t9\n\t\t\t\t\n\t\t\t\n\t\t\t2\n\t\t\n\t\n\t=\n\t16\n\n

","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}