{"id":"cb_question_bank:f38efee1-107c-4b23-acc9-7a96ab256b59","source_id":"cb_question_bank","source_item_id":"f38efee1-107c-4b23-acc9-7a96ab256b59","external_id":"10d4dc39-2d6d-4fb6-8cc4-828292d3cbe0","ibn":null,"question_id":"82c8325f","program":"SAT","module":"math","domain_code":"S","domain":"Geometry and Trigonometry","skill_code":"S.D.","skill":"Circles","difficulty":"M","score_band":5,"answer_type":"mcq","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 4 comma 5 right parenthesis and the point left parenthesis negative 8 comma 8 right parenthesis lies on the circle. Which equation represents this circle?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">A circle in the&nbsp;<em>xy</em>-plane has its center at&nbsp;<math alttext=\"left parenthesis negative 4 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mrow><mn>4</mn></mrow><mo>,</mo><mrow><mn>5</mn></mrow></mrow></mfenced></math> and the point <math alttext=\"left parenthesis negative 8 comma 8 right parenthesis\"><mfenced><mrow><mrow><mo>-</mo><mn>8</mn></mrow><mo>,</mo><mrow><mn>8</mn></mrow></mrow></mfenced></math> lies on the circle. Which equation represents this circle?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. A circle in the <em>xy</em>-plane can be represented by an equation of the form&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&nbsp;<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 the circle and <math alttext=\"r\"><mi>r</mi>\n</math> is the length of a radius of the circle. It's given that the circle has its center at&nbsp;<math alttext=\"left parenthesis negative 4 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math>. Therefore, <math alttext=\"h equals negative 4\"><mrow>\n\t<mi>h</mi>\n\t<mo>=</mo>\n\t<mo>-</mo><mn>4</mn>\n</mrow>\n</math> and <math alttext=\"k equals 5\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>5</mn>\n</mrow>\n</math>. Substituting <math alttext=\"negative 4\"><mo>-</mo><mn>4</mn>\n</math> for <math alttext=\"h\"><mi>h</mi>\n</math> and <math alttext=\"5\"><mn>5</mn>\n</math> for <math alttext=\"k\"><mi>k</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&nbsp;<math alttext=\"left parenthesis x minus left parenthesis negative 4 right parenthesis right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced></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><mi>r</mi><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</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><mi>r</mi><mn>2</mn></msup></math>. It's also given that the point&nbsp;<math alttext=\"left parenthesis negative 8 comma 8 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math> lies on the circle. Substituting <math alttext=\"negative 8\"><mo>-</mo><mn>8</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"8\"><mn>8</mn>\n</math> for <math alttext=\"y\"><mi>y</mi>\n</math> in the equation <math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</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><mi>r</mi><mn>2</mn></msup></math> yields&nbsp;<math alttext=\"left parenthesis negative 8 plus 4 right parenthesis squared plus left parenthesis 8 minus 5 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mo>-</mo><mn>8</mn><mo>+</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mn>8</mn><mo>-</mo><mn>5</mn></mrow></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>, or <math alttext=\"left parenthesis negative 4 right parenthesis squared plus left parenthesis 3 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mn>3</mn></mfenced><mn>2</mn></msup><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>, which is equivalent to&nbsp;<math alttext=\"16 plus 9 equals r squared\"><mn>16</mn><mo>+</mo><mn>9</mn><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>, or&nbsp;<math alttext=\"25 equals r squared\"><mn>25</mn><mo>=</mo><msup><mi>r</mi><mn>2</mn></msup></math>. Substituting <math alttext=\"25\"><mn>25</mn>\n</math> for <math alttext=\"r squared\"><msup>\n\t<mi>r</mi>\n\t<mn>2</mn>\n</msup>\n</math> in the equation&nbsp;<math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals r squared\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</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><mi>r</mi><mn>2</mn></msup></math> yields&nbsp;<math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 25\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</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>25</mn></math>. Thus, the equation&nbsp;<math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 25\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mn>4</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>25</mn></math> represents the circle.</p>\n<p style=\"text-align: left;\">Choice A is incorrect. The circle represented by this equation has its center at&nbsp;<math alttext=\"left parenthesis 4 comma negative 5 right parenthesis\"><mfenced><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math>, not&nbsp;<math alttext=\"left parenthesis negative 4 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math>, and the point&nbsp;<math alttext=\"left parenthesis negative 8 comma 8 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math> doesn't lie on the circle.</p>\n<p style=\"text-align: left;\">Choice B is incorrect. The point&nbsp;<math alttext=\"left parenthesis negative 8 comma 8 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math> doesn't lie on the circle represented by this equation.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. The circle represented by this equation has its center at <math alttext=\"left parenthesis 4 comma negative 5 right parenthesis\"><mfenced><mrow><mn>4</mn><mo>,</mo><mo>-</mo><mn>5</mn></mrow></mfenced></math>, not <math alttext=\"left parenthesis negative 4 comma 5 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></mfenced></math>, and the point&nbsp;<math alttext=\"left parenthesis negative 8 comma 8 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>8</mn><mo>,</mo><mn>8</mn></mrow></mfenced></math> doesn't lie on the circle.</p>","correct_answer":["D"],"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":[{"label":"A","content_html":"<p><math alttext=\"left parenthesis x minus 4 right parenthesis squared plus left parenthesis y plus 5 right parenthesis squared equals 5\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>4</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mrow><mn>5</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>5</mn></math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 5\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mrow><mn>4</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mrow><mn>5</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>5</mn></math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"left parenthesis x minus 4 right parenthesis squared plus left parenthesis y plus 5 right parenthesis squared equals 25\"><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mrow><mn>4</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>+</mo><mrow><mn>5</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>25</mn></math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"left parenthesis x plus 4 right parenthesis squared plus left parenthesis y minus 5 right parenthesis squared equals 25\"><msup><mfenced><mrow><mi>x</mi><mo>+</mo><mrow><mn>4</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>+</mo><msup><mfenced><mrow><mi>y</mi><mo>-</mo><mrow><mn>5</mn></mrow></mrow></mfenced><mn>2</mn></msup><mo>=</mo><mn>25</mn></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}}