{"id":"cb_question_bank:755a182a-7545-4b4c-974e-630747a33794","source_id":"cb_question_bank","source_item_id":"755a182a-7545-4b4c-974e-630747a33794","external_id":null,"ibn":"01735-DC","question_id":"9e44284b","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":"In the xy -plane, the graph of 2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45 is a circle. What is the radius of the circle?","stimulus":"","stimulus_html":null,"stem_html":"<p class=\"stem_paragraph \">In the&nbsp;<span class=\"italic\"><span class=\"italic \">xy</span></span>-plane, the graph of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4b9ace5a783c21f4631f8d7ab0c416b1ae686c47a9ad13e6fd3bdec34b50fed5\" alt=\"2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45\"></span> is a circle. What is the radius of the circle?</p>\n","rationale_html":"<p>Choice A is correct. One way to find the radius of the circle is to rewrite the given equation in standard 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/0fbcdf3067da8003b9a2185c18d77c5f31212a47481a805cefb139af286f21d4\" alt=\"the ordered pair h comma k\"></span> is the center of the circle and the radius of the circle is <span class=\"italic\">r</span>. To do this, divide the original equation, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/4b9ace5a783c21f4631f8d7ab0c416b1ae686c47a9ad13e6fd3bdec34b50fed5\" alt=\"2 x squared, minus 6 x, plus 2 y squared, plus 2 y, equals 45\"></span>, by 2 to make the leading coefficients of <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/3a186314172bc239fb025135d8722d39ee39bfefe4e212836956158aa0b9ff08\" alt=\"x squared\"></span> and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/728c0e4d174d051dae68400947ed6a9b36af1a008ccd441e54e6e1038bfce0f3\" alt=\"y squared\"></span> each equal to 1:&nbsp;<span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/468ef6b8dc2a1401426559192b8df81e5758561e208ea033d7aa70340c8efce9\" alt=\"as follows: x squared, minus 3 x, plus y squared, plus y, equals 22 point 5\"></span>. Then complete the square to put the equation in standard form. To do so, first rewrite <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/468ef6b8dc2a1401426559192b8df81e5758561e208ea033d7aa70340c8efce9\" alt=\"x squared, minus 3 x, plus y squared, plus y, equals 22 point 5\"></span> as <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/40c2c3b968374b345a7da515c302df88f759bd277321a4fef1a9a6ca1c9f16e0\" alt=\"open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, minus 2 point 2 5, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, minus 0 point 2 5, equals 22 point 5\"></span>. Second, add 2.25 and 0.25 to both sides of the equation: <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/688e5edf176753b0084314a7861653fba03dc421cc2fa9aae3bfbd3947ab15e2\" alt=\"open parenthesis, x squared, minus 3 x, plus 2 point 2 5, close parenthesis, plus, open parenthesis, y squared, plus y, plus 0 point 2 5, close parenthesis, equals 25\"></span>. Since <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/ab6908eecdf7ad36ae2e9f28041aee4cba703a86bee46fe9791a6c6941ff118d\" alt=\"x squared, minus 3 x, plus 2 point 2 5, equals, open parenthesis, x minus 1 point 5, close parenthesis, squared\"></span>, <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/9da162663b7ddf169900056f273cf0f6d1c16fd2b5135a7dbe3a892fdd88ad9f\" alt=\"y squared, plus y, plus 0 point 2 5, equals, open parenthesis, y plus 0 point 5, close parenthesis, squared\"></span>, and <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/86a44add3617ec26374b5a9ea7c9440b53b5a7e4810a92c37ae281789377869a\" alt=\"25 equals 5 squared\"></span>, it follows that <span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/902f9f21029a2095f199e4fa74664c996fb821e36e73cebb529c2032dcaed397\" alt=\"open parenthesis, x minus 1 point 5, close parenthesis, squared, plus, open parenthesis, y plus 0 point 5, close parenthesis, squared, equals 5 squared\"></span>. Therefore, the radius of the circle is 5.</p><p>Choices B, C, and D are incorrect and may be the result of errors in manipulating the equation or of a misconception about the standard form of the equation of a circle in the <span class=\"italic\">xy</span>-plane.</p><p>&nbsp;</p><p></p>\n","correct_answer":["A"],"has_media":true,"has_mathml":false,"has_table":false,"source_created_at":"2023-08-02T20:25:59.615000+00:00","source_updated_at":"2023-08-02T20:25:59.615000+00:00","options":[{"label":"A","content_html":"<p>5</p>\n","ord":0},{"label":"B","content_html":"<p>6.5</p>\n","ord":1},{"label":"C","content_html":"<p><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/e5add5f49da72c492311520b9fd297044d5048200609c83edcc0809656af9cd7\" alt=\"square root of 40\"></span></p><p>&nbsp;</p><p></p>\n","ord":2},{"label":"D","content_html":"<p><span class=\"math-container\"><img align=\"middle\" role=\"math\" class=\"math-img\" src=\"/api/assets/6e4d9fc0e97f8af31b7ff4fb59ca6eef4ffd75e442bfc0a5e23c8131d37d93d4\" alt=\"square root of 50\"></span></p><p>&nbsp;</p><p></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}}