{"id":2256,"date":"2024-06-29T10:52:47","date_gmt":"2024-06-29T10:52:47","guid":{"rendered":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/?page_id=2256"},"modified":"2024-06-30T00:11:55","modified_gmt":"2024-06-30T00:11:55","slug":"reflecting-on-ellipses-and-hyperbolas","status":"publish","type":"page","link":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/reflecting-on-ellipses-and-hyperbolas\/","title":{"rendered":"Reflecting on ellipses and hyperbolas"},"content":{"rendered":"\n
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Moroccan Journal of Algebra and Geometry with Applications<\/strong><\/p>\n\n\n\n

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Reflecting on ellipses and hyperbolas<\/strong><\/p>\n<\/div>\n<\/div>\n\n\n\n

David E. Dobbs <\/strong><\/i>
Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA.<\/p>\n\n\n\n

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Pages 115-169 | Received 02 Augus<\/span>t 2023, Accepted 01 November 2023, Published 30 June 2024 <\/span><\/p>\n\n\n\n

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Abstract<\/span><\/strong><\/p>\n\n\n\n

Variants of some known results are proven in this teaching note. Proofs of the reflection property of ellipses and the reflection properties of hyperbolas are given; those proofs would be accessible early in a calculus class or in a course that combines precalculus with an introduction to differential calculus. Also, those reflection properties play key roles in results characterizing ellipses and hyperbolas; their proofs solve initial value problems concerning certain first order ordinary differential equations, and so they would be accessible early in a course on differential equations or in some courses on integral calculus. Special attention is paid to identifying some piecewise linear, not necessarily connected, degenerate cases resulting from proofs of the characterization results.<\/p>\n\n\n\n

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Keywords<\/strong>:<\/span>  Euclidean analytic geometry, ellipse, hyperbola, focus, tangential half-line, vector, dot product, inverse cosine
function, derivative, initial value problem.<\/p>\n\n\n\n

MSC numbers<\/strong>:<\/span> Primary 51-02; Secondary 51N20, 33B10, 97G70, 26A06, 34-01.<\/p>\n\n\n\n

Downloads:<\/strong> Full-text PDF<\/a><\/p>\n<\/div>\n\n\n\n

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Moroccan Journal of Algebra and Geometry with Applications Latest articles Reflecting on ellipses and hyperbolas David E. Dobbs  Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37996-1320, USA. Pages 115-169 | Received 02 August 2023, Accepted 01 November 2023, Published 30 June 2024 Abstract Variants of some known results are proven in this teaching note. Read More<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"template-page-builder.php","meta":{"footnotes":""},"class_list":["post-2256","page","type-page","status-publish","hentry","entry"],"_links":{"self":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2256","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/comments?post=2256"}],"version-history":[{"count":3,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2256\/revisions"}],"predecessor-version":[{"id":2318,"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/pages\/2256\/revisions\/2318"}],"wp:attachment":[{"href":"https:\/\/ced.fst-usmba.ac.ma\/p\/mjaga\/wp-json\/wp\/v2\/media?parent=2256"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}