{"id":"cb_question_bank:42b358f9-789a-4de7-995f-a1c927f41ea9","source_id":"cb_question_bank","source_item_id":"42b358f9-789a-4de7-995f-a1c927f41ea9","external_id":"a8b375f5-371f-458f-bca7-e5b78b360f11","ibn":null,"question_id":"b8a225ff","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"H","score_band":6,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"Circle A in the xy -plane has the equation left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 4 . Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the xy -plane is left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals k , where k is a constant. What is the value of k ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">Circle A in the <em>xy</em>-plane has the equation <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 4\"><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>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</mrow>\n\t<mo>=</mo>\n\t<mn>4</mn>\n</mrow>\n</math>. Circle B has the same center as circle A. The radius of circle B is two times the radius of circle A. The equation defining circle B in the <em>xy</em>-plane is <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals k\"><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>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</mrow>\n\t<mo>=</mo>\n\t<mi>k</mi>\n</mrow>\n</math>, where <math alttext=\"k\"><mi>k</mi>\n</math> is a constant. What is the value of <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p>The correct answer is <math alttext=\"16\"><mn>16</mn>\n</math>. An equation of a circle in the <em>xy</em>-plane can be written as <math alttext=\"left parenthesis x minus t right parenthesis squared plus left parenthesis y minus u right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>u</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>, where the center of the circle is <math alttext=\"left parenthesis t comma u right parenthesis\"><mfenced><mrow><mi>t</mi><mo>,</mo><mi>u</mi></mrow></mfenced></math> , the radius of the circle is <math alttext=\"r\"><mi>r</mi>\n</math>, and where <math alttext=\"t\"><mi>t</mi>\n</math>, <math alttext=\"u\"><mi>u</mi>\n</math>, and <math alttext=\"r\"><mi>r</mi>\n</math> are constants. It’s given that the equation of circle A is <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 4\"><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>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>4</mn></math>, which is equivalent to <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 2 squared\"><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>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>2</mn><mn>2</mn></msup></math>. Therefore, the center of circle A is&nbsp;<math alttext=\"left parenthesis negative 5 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math> and the radius of circle A is <math alttext=\"2\"><mn>2</mn>\n</math>. It’s given that circle B has the same center as circle A and that the radius of circle B is two times the radius of circle A. Therefore, the center of circle B is&nbsp;<math alttext=\"left parenthesis negative 5 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>5</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math> and the radius of circle B is <math alttext=\"2 left parenthesis 2 right parenthesis\"><mn>2</mn><mfenced><mn>2</mn></mfenced></math>, or <math alttext=\"4\"><mn>4</mn>\n</math>. Substituting <math alttext=\"negative 5\"><mo>-</mo><mn>5</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math>, <math alttext=\"5\"><mn>5</mn>\n</math> for <math alttext=\"u\"><mi>u</mi>\n</math>, and <math alttext=\"4\"><mn>4</mn>\n</math> for <math alttext=\"r\"><mi>r</mi>\n</math> into the equation <math alttext=\"left parenthesis x minus t right parenthesis squared plus left parenthesis y minus u right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>t</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mi>u</mi></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>&nbsp; yields&nbsp;<math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 4 squared\"><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>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>4</mn><mn>2</mn></msup></math>, which is equivalent to <math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 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>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>16</mn></math>. It follows that the equation of circle B in the <em>xy</em>-plane is&nbsp;<math alttext=\"left parenthesis x plus 5 right parenthesis squared plus left parenthesis y minus 5 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>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>16</mn></math>. Therefore, the value of <math alttext=\"k\"><mi>k</mi>\n</math> is <math alttext=\"16\"><mn>16</mn>\n</math>.</p>","correct_answer":["16"],"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}}