{"id":"cb_question_bank:37cd15af-2ad4-43d3-938e-94b4d86613d5","source_id":"cb_question_bank","source_item_id":"37cd15af-2ad4-43d3-938e-94b4d86613d5","external_id":null,"ibn":"08283-DC","question_id":"2266984b","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":"The equation above defines a circle in the xy -plane. What are the coordinates of the center of the circle?","stimulus":"x squared, plus 20 x, plus y squared, plus 16 y, equals negative 20","stimulus_html":"<div class=\"stimulus_reference \">\n\t<p class=\"standalone_statement style:1 \"><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/bcec2d9cdd050066eff45b1e2871841ab2b004801f1d5d67b1ac1d4b29935880\" alt=\"x squared, plus 20 x, plus y squared, plus 16 y, equals negative 20\"></span></span></p>\n</div>\n","stem_html":"<p class=\"stem_paragraph \">The equation above defines a circle in the <span class=\"italic\"><span class=\"italic \">xy</span></span>-plane. What are the coordinates of the center of the circle?</p>\n","rationale_html":"<p>Choice B is correct. The standard equation of a circle in the <span class=\"italic\">xy</span>-plane is of the form <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d573a336dc1753ebd43a919d2d966269eabeead14b993a357712d5d7ea39eac9\" alt=\"open parenthesis, x minus h, close parenthesis, squared, plus, open parenthesis, y minus k, close parenthesis, squared, equals r squared\"></span>, where <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/c55f293830481ca9be1517e3fda14d86e6cdc21597a1ea3c45b3e1fd7649e99c\" alt=\"the ordered pair h comma k\"></span>&nbsp;are the coordinates of the center of the circle and <span class=\"italic\">r</span> is the radius. The given equation can be rewritten in standard form by completing the squares. So the sum of the first two terms, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d3936e5070a7bc1ef4739633f18f283a7c3f94269bad8430648c8d176a37ca0f\" alt=\"x squared, plus 20 x\"></span>, needs a 100 to complete the square, and the sum of the second two terms, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9583dda90442e3c6bc0afc2cd1f9b5379b5da4922f30e02dd62744cc987842e9\" alt=\"y squared, plus 16 y\"></span>, needs a 64 to complete the square. Adding 100 and 64 to both sides of the given equation yields <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/b2151d1a6dcdbc69e40c63214f72a5952e15361b197834ae20b565cc4674ef79\" alt=\"open parenthesis, x squared, plus 20 x, plus 100, close parenthesis, plus, open parenthesis, y squared, plus 16 y, plus 64, close parenthesis, equals negative 20, plus 100, plus 64\"></span>, which is equivalent to <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/f97588fb40b10f191015b15a2a9a392fb0cdb740b21202b857c6797cec01187f\" alt=\"open parenthesis, x plus 10, close parenthesis, squared, plus, open parenthesis, y plus 8, close parenthesis, squared, equals 144\"></span>. Therefore, the coordinates of the center of the circle are <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/44221392b69e1f294c2eef5b808f2680c0a5d7e04dd2e617f23a5c2c559c6035\" alt=\"negative 10 comma negative 8\"></span>.</p><p>Choices A, C, and D are incorrect and may result from computational errors made when attempting to complete the squares or when identifying the coordinates of the center.</p><p>&nbsp;</p><p></p>\n","correct_answer":["B"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.635000+00:00","source_updated_at":"2023-08-02T20:25:59.635000+00:00","options":[{"label":"A","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/20720dec09db9a169f5114dc36c7e32aeb2a6fa3fa6ecb72285e73f83d57c0f5\" alt=\"negative 20 comma negative 16\"></span></span></p>\n","ord":0},{"label":"B","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/d1af191438c62573730955697e52605bf7b9e0d266a6ef789033baad79e7d75b\" alt=\"negative 10 comma negative 8\"></span></span></p>\n","ord":1},{"label":"C","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3b8cb7b434794db1f27aa5a6dc3813fdde9f0ce1566294aefb698113345ab1bc\" alt=\"10 comma 8\"></span></span></p>\n","ord":2},{"label":"D","content_html":"<p><span class=\"math_expression \"><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/34dc69845534fbba7e3c3bd8313f4221c24181d58424e126556a8ad348d8b7da\" alt=\"20 comma 16\"></span></span></p>\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}}