{"id":"cb_question_bank:b06f5441-56db-438e-8689-7b73ffdd0cff","source_id":"cb_question_bank","source_item_id":"b06f5441-56db-438e-8689-7b73ffdd0cff","external_id":"51707cf8-d699-4a87-a4c7-1cf41c8e4d28","ibn":null,"question_id":"89661424","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"A circle in the xy -plane has its center at left parenthesis negative 5 comma 2 right parenthesis and has a radius of 9 . An equation of this circle is x squared plus y squared plus a x plus b y plus c equals 0 , where a , b , and c are constants. What is the value of c ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A circle in the <em>xy</em>-plane has its center at <math alttext=\"left parenthesis negative 5 comma 2 right parenthesis\"><mfenced><mrow><mrow><mo>-</mo><mn>5</mn></mrow><mo>,</mo><mrow><mn>2</mn></mrow></mrow></mfenced></math> and has a radius of <math alttext=\"9\"><mn>9</mn>\n</math>. An equation of this circle is <math alttext=\"x squared plus y squared plus a x plus b y plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants. What is the value of <math alttext=\"c\"><mi>c</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 52\"><mo>-</mo><mn>52</mn>\n</math>. The equation of a circle in the <em>xy</em>-plane with its center at&nbsp;<math alttext=\"left parenthesis h comma k right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math> and a radius of <math alttext=\"r\"><mi>r</mi>\n</math> can be written in the form <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>. It's given that a circle in the <em>xy</em>-plane has its center at&nbsp;<math alttext=\"left parenthesis negative 5 comma 2 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>2</mn></mrow></mfenced></math> and has a radius of <math alttext=\"9\"><mn>9</mn>\n</math>. Substituting <math alttext=\"negative 5\"><mo>-</mo><mn>5</mn>\n</math> for <math alttext=\"h\"><mi>h</mi>\n</math>, <math alttext=\"2\"><mn>2</mn>\n</math> for <math alttext=\"k\"><mi>k</mi>\n</math>, and <math alttext=\"9\"><mn>9</mn>\n</math> for <math alttext=\"r\"><mi>r</mi>\n</math> in the equation&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> yields <math alttext=\"left parenthesis x minus left parenthesis negative 5 right parenthesis right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 9 squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfenced><mrow><mo>-</mo><mn>5</mn></mrow></mfenced></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>9</mn><mn>2</mn></msup></math>, or <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 81\"><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>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>81</mn></math>. It's also given that an equation of this circle is <math alttext=\"x squared plus y squared plus a x plus b y plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"b\"><mi>b</mi>\n</math>, and <math alttext=\"c\"><mi>c</mi>\n</math> are constants. Therefore,&nbsp;<math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 81\"><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>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>81</mn></math> can be rewritten in the form <math alttext=\"x squared plus y squared plus a x plus b y plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>. The equation <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 2 right parenthesis squared equals 81\"><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>2</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>81</mn></math>, or <math alttext=\"left parenthesis x plus 5 right parenthesis left parenthesis x plus 5 right parenthesis plus left parenthesis y minus 2 right parenthesis left parenthesis y minus 2 right parenthesis equals 81\"><mfenced><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>5</mn></mrow></mfenced><mo>+</mo><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced><mo>=</mo><mn>81</mn></math>, can be rewritten as <math alttext=\"x squared plus 5 x plus 5 x plus 25 plus y squared minus 2 y minus 2 y plus 4 equals 81\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>5</mn><mi>x</mi><mo>+</mo><mn>25</mn><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>y</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>+</mo><mn>4</mn><mo>=</mo><mn>81</mn></math>. Combining like terms on the left-hand side of this equation yields <math alttext=\"x squared plus y squared plus 10 x minus 4 y plus 29 equals 81\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>-</mo><mn>4</mn><mi>y</mi><mo>+</mo><mn>29</mn><mo>=</mo><mn>81</mn></math>. Subtracting <math alttext=\"81\"><mn>81</mn>\n</math> from both sides of this equation yields <math alttext=\"x squared plus y squared plus 10 x minus 4 y minus 52 equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>-</mo><mn>4</mn><mi>y</mi><mo>-</mo><mn>52</mn><mo>=</mo><mn>0</mn></math>, which is equivalent to&nbsp;<math alttext=\"x squared plus y squared plus 10 x plus left parenthesis negative 4 right parenthesis y plus left parenthesis negative 52 right parenthesis equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mn>10</mn><mi>x</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mi>y</mi><mo>+</mo><mfenced><mrow><mo>-</mo><mn>52</mn></mrow></mfenced><mo>=</mo><mn>0</mn></math>. This equation is in the form <math alttext=\"x squared plus y squared plus a x plus b y plus c equals 0\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mi>y</mi><mo>+</mo><mi>c</mi><mo>=</mo><mn>0</mn></math>. Therefore, the value of <math alttext=\"c\"><mi>c</mi>\n</math> is <math alttext=\"negative 52\"><mo>-</mo><mn>52</mn>\n</math>.</p>","correct_answer":["-52"],"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":[],"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}}