In late 1941, in an oral exam, my distinguished Professor Don Raul Valdivieso, raised the following issue: <\/p>\n
\"Find the locus of the centre of an ellipse that moves tangent to the positive x- and y-axes.\" <\/p>\n
After brief reflection, I remembered a problem posed in Course Geometry Plana<\/em> (year 1940) by the eminent Professor Don Luciano Claude which read as follows: <\/p>\n \"Find the locus of the point where two straight orthogonal lines intersect, and which are tangential to a given ellipse.\" <\/p>\n The solution to this problem, easy to find in any treaty on conics, is a concentric circle to an ellipse given with the radius equal to: <\/p>\n √(a2<\/sup> + b2<\/sup>), where a and b are the semi-axis of ellipse.\n<\/p>\n<\/blockquote>\n Explanation:<\/strong> \"Orthogonal\" means \"at right angles\" and as we move the pair of orthogonal lines around the ellipse so that they touch the ellipse at one point (they are tangents), a circle is formed by the intersection point: <\/p>\n Here's the resulting circle, which is concentric with the ellipse (this means it has the same centre). <\/p>\n Back to Raul's story:<\/strong><\/p>\n So it is easy to deduce that the inverse of this problem is to consider the orthogonal straight tangents as fixed<\/strong> and the ellipse given as mobile<\/strong>, returning to the problem of Professor Valdivieso. <\/p>\n Clearly the locus will be a portion of the arc of a circle.<\/p>\n However, Professor Valdivieso, even accepting that this was a good solution, required me to follow the long road, namely the analytic geometric deduction directly as initialy requested.<\/p>\n Result:<\/strong> I was not able to find such a solution.<\/p>\n The years passed and my attempts were in vain to find a solution following the method that I had being requested. Several colleagues whom I consulted could not resolve it.<\/p>\n So, already retired and close to fulfilling 85 years, I went back to this issue. The Google Scholar<\/strong> was no help to find what I was are looking for. Finally, using the Central Library at the Catholic University and recalling those old fragile, but extraordinary French books, I found what I was looking for in Lecons de G\u00c3\u00a9om\u00c3\u00a9trie analytique<\/em>. Author: Charles Roguet. Edition year 1860. Page 218. <\/p>\n We need to resolve the previously stated locus,to develop:<\/p>\n SECTION (A)<\/strong> the equation of a tangent line to the ellipse and <\/p>\n SECTION (B)<\/strong> The equation of the distance from a given point to a given straight line. <\/p>\n<\/blockquote>\n What follows is quite a lengthy analysis using the characteristics of conic sections<\/a>.<\/p>\n So dear Reader, can you solve it given the hints above?<\/p>\n The final result is the red arc that you see in the following diagram (the yellow and green curves are not relevant to this problem, even though they are interesting):<\/p>\n Image source: MathCurve.<\/p>\n Raul finished his note as follows<\/strong>, with a challenge to find a simpler solution than his 4 pages of algebra:<\/p>\n That was it. To this day I can not explain that Professor Raul Valdivieso expected that any students could develop this problem, in an examination or in any other circumstance. <\/p>\n It there a much simpler solution? I venture to think there has to be one, because the distinguished professor was very demanding but very thoughtful and fair. <\/p>\n Ra\u00c3\u00bal Espinosa Wellmann, Civil Engineer, Pontificia Universidad Cat\u00c3\u00b3lica de Chile. <\/p>\n<\/blockquote>\n Thanks, Raul for the interesting story and for the evidence that a well-crafted problem is simple to state, and motivating for years.<\/p>\n <\/p>\n Update:<\/b> A reader asked for the full solution, and so here you go. I think you'll agree it is quite an extraordinary question for an examination!<\/p>\n Locus of an ellipse<\/a> [MS Word document]<\/p>\n See the 3 Comments<\/a> below.<\/p>\n","protected":false},"excerpt":{"rendered":"![\"ellipse1\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse1.gif\")
![\"ellipse2\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse2.gif\")
![\"ellipse3\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse3.gif\")
![\"ellipse4\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse4.gif\")
<\/p>\n![\"ellipse5\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse5.gif\")
<\/p>\n\n
![\"glissetteellipse\"](\"\/blog\/wp-content\/images\/2008\/06\/glissetteellipse.gif\")
<\/p>\n\n
![\"ellipse0\"](\"\/blog\/wp-content\/images\/2008\/06\/ellipse0.gif\")
<\/a>A reader from Chile provides an explanation of the locus of the center of an ellipse which moves such that it is tangent to the positive x- and y-axes.<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_mo_disable_npp":""},"categories":[4],"tags":[134,127],"_links":{"self":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/1243"}],"collection":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/comments?post=1243"}],"version-history":[{"count":0,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/posts\/1243\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/media?parent=1243"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/categories?post=1243"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.intmath.com\/blog\/wp-json\/wp\/v2\/tags?post=1243"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}