{"id":"cb_question_bank:3254b028-4bc1-4a5d-a56e-32a25fd70df6","source_id":"cb_question_bank","source_item_id":"3254b028-4bc1-4a5d-a56e-32a25fd70df6","external_id":"301edaf1-d774-4eda-96b8-80c08070887f","ibn":null,"question_id":"76c73dbf","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":"The graph of x squared plus x plus y squared plus y equals StartFraction 199 Over 2 EndFraction in the xy -plane is a circle. What is the length of the circle’s radius?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The graph of <math alttext=\"x squared plus x plus y squared plus y equals StartFraction 199 Over 2 EndFraction\"><mrow>\n\t<mrow>\n\t\t<msup>\n\t\t\t<mi>x</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>+</mo>\n\t\t<mi>x</mi>\n\t\t<mo>+</mo>\n\t\t<msup>\n\t\t\t<mi>y</mi>\n\t\t\t<mn>2</mn>\n\t\t</msup>\n\t\t<mo>+</mo>\n\t\t<mi>y</mi>\n\t</mrow>\n\t<mo>=</mo>\n\t<mfrac>\n\t\t<mn>199</mn>\n\t\t<mn>2</mn>\n\t</mfrac>\n</mrow>\n</math> in the <em>xy</em>-plane is a circle. What is the length of the circle’s radius?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"10\"><mn>10</mn>\n</math>. It's given that the graph of&nbsp;<math alttext=\"x squared plus x plus y squared plus y equals StartFraction 199 Over 2 EndFraction\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac></math> in the <em>xy</em>-plane is a circle. The equation of a circle in the <em>xy</em>-plane 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>, 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>. The term <math alttext=\"left parenthesis x minus h right parenthesis squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup></math> in this equation can be obtained by adding the square of half the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> to both sides of the given equation to complete the square. The coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> is <math alttext=\"1\"><mn>1</mn>\n</math>. Half the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> is <math alttext=\"one half\"><mfrac>\n\t<mn>1</mn>\n\t<mn>2</mn>\n</mfrac>\n</math>. The square of half the coefficient of <math alttext=\"x\"><mi>x</mi>\n</math> is <math alttext=\"one fourth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n</math>. Adding <math alttext=\"one fourth\"><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n</math> to each side of <math alttext=\"left parenthesis x squared plus x right parenthesis plus left parenthesis y squared plus y right parenthesis equals StartFraction 199 Over 2 EndFraction\"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac></math> yields&nbsp;<math alttext=\"left parenthesis x squared plus x plus one fourth right parenthesis plus left parenthesis y squared plus y right parenthesis equals StartFraction 199 Over 2 EndFraction plus one fourth\"><mfenced><mrow><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo>+</mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>, or <math alttext=\"left parenthesis x plus one half right parenthesis squared plus left parenthesis y squared plus y right parenthesis equals StartFraction 199 Over 2 EndFraction plus one fourth\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>. Similarly, the term <math alttext=\"left parenthesis y minus k right parenthesis squared\"><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>k</mi></mrow></mfenced><mn>2</mn></msup></math> can be obtained by adding the square of half the coefficient of <math alttext=\"y\"><mi>y</mi>\n</math> to both sides of this equation, which yields <math alttext=\"left parenthesis x plus one half right parenthesis squared plus left parenthesis y squared plus y plus one fourth right parenthesis equals StartFraction 199 Over 2 EndFraction plus one fourth plus one fourth\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mfenced><mrow><msup><mi>y</mi><mn>2</mn></msup><mo>+</mo><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></mrow></mfenced><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>, or <math alttext=\"left parenthesis x plus one half right parenthesis squared plus left parenthesis y plus one half right parenthesis squared equals StartFraction 199 Over 2 EndFraction plus one fourth plus one fourth\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mfrac><mn>199</mn><mn>2</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac></math>. This equation is equivalent to <math alttext=\"left parenthesis x plus one half right parenthesis squared plus left parenthesis y plus one half right parenthesis squared equals 100\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>100</mn></math>, or&nbsp;<math alttext=\"left parenthesis x plus one half right parenthesis squared plus left parenthesis y plus one half right parenthesis squared equals 10 squared\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>10</mn><mn>2</mn></msup></math>. Therefore, the length of the circle's radius is <math alttext=\"10\"><mn>10</mn>\n</math>.</p>","correct_answer":["10"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.841000+00:00","source_updated_at":"2023-08-02T20:25:59.841000+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}}