{"id":3175,"date":"2023-05-13T13:13:16","date_gmt":"2023-05-13T17:13:16","guid":{"rendered":"https:\/\/josmfs.net\/?p=3175"},"modified":"2024-04-05T16:23:41","modified_gmt":"2024-04-05T20:23:41","slug":"putnam-ellipse-areas-problem","status":"publish","type":"post","link":"https:\/\/josmfs.net\/wordpress\/2023\/05\/13\/putnam-ellipse-areas-problem\/","title":{"rendered":"Putnam Ellipse Areas Problem"},"content":{"rendered":"<p><img loading=\"lazy\" decoding=\"async\" class=\"alignleft  wp-image-3173\" src=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/05\/Putnam-Ellipse-Areas-Fig2.jpg\" alt=\"\" width=\"200\" height=\"204\" srcset=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/05\/Putnam-Ellipse-Areas-Fig2.jpg 400w, https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/05\/Putnam-Ellipse-Areas-Fig2-295x300.jpg 295w\" sizes=\"auto, (max-width: 200px) 100vw, 200px\" \/>This is a nifty <a href=\"https:\/\/mindyourdecisions.com\/blog\/2023\/04\/24\/putnam-ellipse-areas-question\/\">problem <\/a>from Presh Talwakar.<\/p>\n<p>\u201cThis is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion!<\/p>\n<p>Let <em>R<\/em> be the region in the first quadrant bounded by the <em>x<\/em>-axis, the line <em>y<\/em> = <em>x<\/em>\/2, and the ellipse <em>x<\/em><sup>2<\/sup>\/9 + <em>y<\/em><sup>2<\/sup> = 1. Let <em>R<\/em>&#8216; be the region in the first quadrant bounded by the <em>y<\/em>-axis, the line <em>y<\/em> = <em>mx<\/em> and the ellipse. Find the value of <em>m<\/em> such that <em>R<\/em> and <em>R<\/em>&#8216; have the same area.\u201d<\/p>\n<p><a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2024\/04\/Putnam-Ellipse-Areas-Answer.jpg\">Answer<\/a>.<\/p>\n<p>See the <a href=\"https:\/\/josmfs.net\/wordpress\/wp-content\/uploads\/2023\/05\/Putnam-Ellipse-Areas-230513.pdf\">Putnam Ellipse Areas Problem<\/a> for solution.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>This is a nifty problem from Presh Talwakar. \u201cThis is adapted from the 1994 Putnam, A2. Thanks to Nirman for the suggestion! Let R be the region in the first quadrant bounded by the x-axis, the line y = x\/2, and the ellipse x2\/9 + y2 = 1. Let R&#8216; be the region in the [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[3],"tags":[13,17,131],"class_list":["post-3175","post","type-post","status-publish","format-standard","hentry","category-puzzles-and-problems","tag-plane-geometry","tag-presh-talwalkar","tag-putnam-exam"],"_links":{"self":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3175","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/comments?post=3175"}],"version-history":[{"count":2,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3175\/revisions"}],"predecessor-version":[{"id":3543,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/posts\/3175\/revisions\/3543"}],"wp:attachment":[{"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/media?parent=3175"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/categories?post=3175"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/josmfs.net\/wordpress\/wp-json\/wp\/v2\/tags?post=3175"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}