{"title":"CONIC SECTIONS","authors":"David Pierce","doi":"10.1142/9789814295871_0005","DOIUrl":"https://doi.org/10.1142/9789814295871_0005","url":null,"abstract":"A straight line is the shortest path. 1. Ant Z is thirsty and tired. He wants to visit a straight stream nearby on his way home. Find the shortest path for him. (Given two points A,B and a line l not passing through A and B. Find a point X on l such that AX + XB is the minimal possible.) 2. Ant Z is sitting at a vertex of a cubic box. He wants to get to the opposite vertex. Find the shortest part for him on the surface of the box. 3. Find the shortest path for Ant Z: (a) from a point A to a point B both on a side of a cylindrical can; (b) from a point A to a point B on an ice-cream cone; (c) from a point A to a point B on a ball. 4. There are two balls, red and white, on the rectangular billiard table. You want to strike the red ball so that it hits two walls of the table first and then hits the white ball. How should you direct the red ball? Ellipse. Fix two points A and B and the number a > AB. The set of all points X on the plane such that AX + XB = a is called an ellipse. The points A and B are called the foci (plural of focus) of an ellipse. The distance between the foci of an ellipse is called the focal distance. In case when two foci coincide an ellipse becomes a circle. The center of an ellipse is the midpoint of the segment joining its foci. A diameter is a chord passing through the center. All planets’ orbits are ellipses with the sun at a focus. 5. An ellipse which is not a circle has two axes of symmetry: one is the line AB and another one is the perpendicular bisector to AB. The segments inside the ellipse cut by those lines are called the major axis and the minor axis. 6. Let a be the major axis, b be the minor axis and c be the focal distance. Show that c = a − b. A tangent line to an ellipse is a line that intersects the ellipse at exactly one point. For any point X on an ellipse there exists a unique line through X tangent to the ellipse. 7. Let X be a point of an ellipse Γ, t be the tangent line to Γ at X. Let A and B be two foci of Γ. Show that the angle between AX and t equals the angle between BX and t.","PeriodicalId":125561,"journal":{"name":"Modern Astrodynamics","volume":"295 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131851924","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"THE f AND g FUNCTIONS","authors":"","doi":"10.2307/j.ctv18zhf27.8","DOIUrl":"https://doi.org/10.2307/j.ctv18zhf27.8","url":null,"abstract":"","PeriodicalId":125561,"journal":{"name":"Modern Astrodynamics","volume":"98 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133709059","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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