{"id":"cb_question_bank:43d0f475-1558-44cb-803a-d8b8272cd5b7","source_id":"cb_question_bank","source_item_id":"43d0f475-1558-44cb-803a-d8b8272cd5b7","external_id":"20c5df6f-78ac-41d4-99b0-3fda967fa4b0","ibn":null,"question_id":"b0a72bdc","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"What is the diameter of the circle in the xy -plane with equation left parenthesis x minus 5 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 16 ?","stimulus":"","stimulus_html":null,"stem_html":"<p>What is the diameter of the circle in the <em>xy</em>-plane with equation <math alttext=\"left parenthesis x minus 5 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 16\"><mrow>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mfenced>\n\t\t\t\t<mrow>\n\t\t\t\t\t<mi>x</mi>\n\t\t\t\t\t<mo>-</mo>\n\t\t\t\t\t<mn>5</mn>\n\t\t\t\t</mrow>\n\t\t\t</mfenced>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>+</mo>\n\t\t<msup>\n\t\t\t<mfenced>\n\t\t\t\t<mrow>\n\t\t\t\t\t<mi>y</mi>\n\t\t\t\t\t<mo>-</mo>\n\t\t\t\t\t<mn>3</mn>\n\t\t\t\t</mrow>\n\t\t\t</mfenced>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t</mrow>\n\t<mo>=</mo>\n\t<mn>16</mn>\n</mrow>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice B is correct. The standard form of an equation of a circle in the <em>xy</em>-plane is&nbsp;<math alttext=\"left parenthesis x minus h right parenthesis squared plus left parenthesis y minus k right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>, where the coordinates of the center of the circle are&nbsp;<math alttext=\"left parenthesis h comma k right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math> and the length of the radius of the circle is <math alttext=\"r\"><mi>r</mi>\n</math>. For the circle in the <em>xy</em>-plane with equation <math alttext=\"left parenthesis x minus 5 right parenthesis squared plus left parenthesis y minus 3 right parenthesis squared equals 16\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>3</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>16</mn></math>, it follows that <math alttext=\"r squared equals 16\"><msup><mi>r</mi><mn>2</mn></msup><mo>=</mo><mn>16</mn></math>. Taking the square root of both sides of this equation yields <math alttext=\"r equals 4\"><mrow>\n\t<mi>r</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math> or <math alttext=\"r equals negative 4\"><mrow>\n\t<mi>r</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>4</mn>\n</mrow>\n</math>. Because <math alttext=\"r\"><mi>r</mi>\n</math> represents the length of the radius of the circle and this length must be positive, <math alttext=\"r equals 4\"><mrow>\n\t<mi>r</mi>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Therefore, the radius of the circle is <math alttext=\"4\"><mn>4</mn>\n</math>. The diameter of a circle is twice the length of the radius of the circle. Thus, <math alttext=\"2 left parenthesis 4 right parenthesis\"><mn>2</mn><mfenced><mn>4</mn></mfenced></math> yields <math alttext=\"8\"><mn>8</mn>\n</math>. Therefore, the diameter of the circle is <math alttext=\"8\"><mn>8</mn>\n</math>.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. This is the radius of the circle.&nbsp;</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This is the square of the radius of the circle.&nbsp;</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["B"],"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":"<p><math alttext=\"4\"><mn>4</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"8\"><mn>8</mn>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"16\"><mn>16</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"32\"><mn>32</mn>\n</math></p>","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}}