{"id":"cb_question_bank:9852356b-bf9b-48be-ae51-b7d6e192c081","source_id":"cb_question_bank","source_item_id":"9852356b-bf9b-48be-ae51-b7d6e192c081","external_id":"a786b147-72b1-4d1e-b373-354a8186f3a8","ibn":null,"question_id":"9acd101f","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":"The equation x squared plus left parenthesis y minus 1 right parenthesis squared equals 49 represents circle A. Circle B is obtained by shifting circle A down 2 units in the xy -plane. Which of the following equations represents circle B?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The equation <math alttext=\"x squared plus left parenthesis y minus 1 right parenthesis squared equals 49\"><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<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>1</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>49</mn>\n</mrow>\n</math> represents circle A. Circle B is obtained by shifting circle A down <math alttext=\"2\"><mn>2</mn>\n</math> units in the <em>xy</em>-plane. Which of the following equations represents circle B?</p>","rationale_html":"<p>Choice D is correct. The graph in the <em>xy</em>-plane of an equation of 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> is a circle with center&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 length <math alttext=\"r\"><mi>r</mi>\n</math>. It's given that circle A is represented by <math alttext=\"x squared plus left parenthesis y minus 1 right parenthesis squared equals 49\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>49</mn></math>, which can be rewritten as <math alttext=\"x squared plus left parenthesis y minus 1 right parenthesis squared equals 7 squared\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>7</mn><mn>2</mn></msup></math>. Therefore, circle A has center&nbsp;<math alttext=\"left parenthesis 0 comma 1 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></mfenced></math> and a radius of length <math alttext=\"7\"><mn>7</mn>\n</math>. Shifting circle A down two units is a rigid vertical translation of circle A that does not change its size or shape. Since circle B is obtained by shifting circle A down two units, it follows that circle B has the same radius as circle A, and for each point <math alttext=\"left parenthesis x comma y right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></mfenced></math> on circle A, the point <math alttext=\"left parenthesis x comma y minus 2 right parenthesis\"><mfenced><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> lies on circle B. Moreover, if <math alttext=\"left parenthesis h comma k right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math> is the center of circle A, then <math alttext=\"left parenthesis h comma k minus 2 right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi><mo>-</mo><mn>2</mn></mrow></mfenced></math> is the center of circle B. Therefore, circle B has a radius of <math alttext=\"7\"><mn>7</mn>\n</math> and the center of circle B is <math alttext=\"left parenthesis 0 comma 1 minus 2 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>1</mn><mo>-</mo><mn>2</mn></mrow></mfenced></math>, or <math alttext=\"left parenthesis 0 comma negative 1 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>1</mn></mrow></mfenced></math>. Thus, circle B can be represented by the equation <math alttext=\"x squared plus left parenthesis y plus 1 right parenthesis squared equals 7 squared\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mn>7</mn><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"x squared plus left parenthesis y plus 1 right parenthesis squared equals 49\"><msup><mi>x</mi><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>49</mn></math>.</p>\n<p>Choice A is incorrect. This is the equation of a circle obtained by shifting circle A right <math alttext=\"2\"><mn>2</mn>\n</math> units.</p>\n<p>Choice B is incorrect. This is the equation of a circle obtained by shifting circle A up <math alttext=\"2\"><mn>2</mn>\n</math> units.</p>\n<p>Choice C is incorrect. This is the equation of a circle obtained by shifting circle A left <math alttext=\"2\"><mn>2</mn>\n</math> units.</p>","correct_answer":["D"],"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=\"left parenthesis x minus 2 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 49\"><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>2</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>1</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>49</mn>\n</mrow>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"x squared plus left parenthesis y minus 3 right parenthesis squared equals 49\"><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<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>49</mn>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis x plus 2 right parenthesis squared plus left parenthesis y minus 1 right parenthesis squared equals 49\"><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>2</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>1</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>49</mn>\n</mrow>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"x squared plus left parenthesis y plus 1 right parenthesis squared equals 49\"><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<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>1</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>49</mn>\n</mrow>\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}}