{"id":"cb_question_bank:60c77f02-3298-4f35-8572-7c9b1eb475e4","source_id":"cb_question_bank","source_item_id":"60c77f02-3298-4f35-8572-7c9b1eb475e4","external_id":"bf1e177f-7c4f-4d1a-9d5d-36f0191c1cde","ibn":null,"question_id":"3e577e4a","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":"A circle in the xy -plane has its center at left parenthesis negative 4 comma negative 6 right parenthesis . Line k is tangent to this circle at the point left parenthesis negative 7 comma negative 7 right parenthesis . What is the slope of line k ?","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 4 comma negative 6 right parenthesis\"><mfenced><mrow><mrow><mo>-</mo><mn>4</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>6</mn></mrow></mrow></mfenced></math>. Line <math alttext=\"k\"><mi>k</mi>\n</math> is tangent to this circle at the point <math alttext=\"left parenthesis negative 7 comma negative 7 right parenthesis\"><mfenced><mrow><mrow><mo>-</mo><mn>7</mn></mrow><mo>,</mo><mrow><mo>-</mo><mn>7</mn></mrow></mrow></mfenced></math>. What is the slope of line <math alttext=\"k\"><mi>k</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. A line that's tangent to a circle is perpendicular to the radius of the circle at the point of tangency. It's given that the circle has its center at <math alttext=\"left parenthesis negative 4 comma negative 6 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>6</mn></mrow></mfenced></math> and line <math alttext=\"k\"><mi>k</mi>\n</math> is tangent to the circle at the point <math alttext=\"left parenthesis negative 7 comma negative 7 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mn>7</mn></mrow></mfenced></math>. The slope of a radius defined by the points <math alttext=\"left parenthesis q comma r right parenthesis\"><mfenced><mrow><mi>q</mi><mo>,</mo><mi>r</mi></mrow></mfenced></math> and <math alttext=\"left parenthesis s comma t right parenthesis\"><mfenced><mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></mfenced></math> can be calculated as <math alttext=\"StartFraction t minus r Over s minus q EndFraction\"><mfrac><mrow><mi>t</mi><mo>-</mo><mi>r</mi></mrow><mrow><mi>s</mi><mo>-</mo><mi>q</mi></mrow></mfrac></math>. The points <math alttext=\"left parenthesis negative 7 comma negative 7 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mn>7</mn></mrow></mfenced></math> and <math alttext=\"left parenthesis negative 4 comma negative 6 right parenthesis\"><mfenced><mrow><mo>-</mo><mn>4</mn><mo>,</mo><mo>-</mo><mn>6</mn></mrow></mfenced></math> define the radius of the circle at the point of tangency. Therefore, the slope of this radius can be calculated as <math alttext=\"StartFraction left parenthesis negative 6 right parenthesis minus left parenthesis negative 7 right parenthesis Over left parenthesis negative 4 right parenthesis minus left parenthesis negative 7 right parenthesis EndFraction\"><mfrac><mrow><mfenced><mrow><mo>-</mo><mn>6</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>7</mn></mrow></mfenced></mrow><mrow><mfenced><mrow><mo>-</mo><mn>4</mn></mrow></mfenced><mo>-</mo><mfenced><mrow><mo>-</mo><mn>7</mn></mrow></mfenced></mrow></mfrac></math>, or <math alttext=\"one third\"><mfrac><mn>1</mn><mn>3</mn></mfrac></math>. If a line and a radius are perpendicular, the slope of the line must be the negative reciprocal of the slope of the radius. The negative reciprocal of <math alttext=\"one third\"><mfrac><mn>1</mn><mn>3</mn></mfrac></math> is <math alttext=\"negative 3\"><mo>-</mo><mn>3</mn></math>. Thus, the slope of line <math alttext=\"k\"><mi>k</mi>\n</math> is&nbsp;<math alttext=\"negative 3\"><mo>-</mo><mn>3</mn></math>.</p>\n<p style=\"text-align: left;\">Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p style=\"text-align: left;\">Choice C is incorrect. This is the slope of the radius of the circle at the point of tangency, not the slope of line <math alttext=\"k\"><mi>k</mi>\n</math>.</p>\n<p style=\"text-align: left;\">Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"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=\"negative 3\"><mo>-</mo><mn>3</mn>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math alttext=\"negative one third\"><mrow>\n\t<mo>-</mo>\n\t<mfrac>\n\t\t<mn>1</mn>\n\t\t<mn>3</mn>\n\t</mfrac>\n</mrow>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math alttext=\"one third\"><mfrac>\n\t<mn>1</mn>\n\t<mn>3</mn>\n</mfrac>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math alttext=\"3\"><mn>3</mn>\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}}