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is the focus question of the extended essay?","Question",{"text":63,"@type":64},"It asks: on a circular billiards table with two balls, at what point along the circumference must one aim so the ball strikes the other after rebounding off the edge.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"What methods are used to solve the problem?",{"text":68,"@type":64},"The essay uses coordinate geometry and trigonometric principles, first trying an initial approach with chord measurements, then analyzing special cases and forming a general solution.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the final solution verified?",{"text":72,"@type":64},"The solution is checked through specific examples using technology such as Autograph software and a TI-84 graphing 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\nAntwerp International SchoolMay 2007  \nWord Count:3017  \nAbstract  \nThe research question of this Mathematics Extended Essay is,\"on a circular tablethere are two balls;at what point along the circumference must one be aimed at in orderfor it to strike the other after rebounding off the edge”.In investigating this question,Ifirst used my own initial approach(which involved measuring various chord lengths),followed by looking at a number of special cases scenarios(for example when both ballsare on the diameter,or equidistant from the center)and finally forming a general solutionbased on coordinate geometry and trigonometric principles.The investigation includedusing an idea provided by Heinrich Dorrie and with the use of diagrams and a lengthymathematical analysis with a large emphasis on trigonometric identities,a solution wasfound.The conclusion reached is,\"if we are given the coordinate plane positions ofbilliard ball A with coordinates(xA,VA)and billiard ball B with coordinates(xB,YB),andalso the radius of the circle,the solution points are at any of the points of intersection ofthe circular table with the hyperbola,(x²-y²)P+(r²)(yp-xm)+(xy2M)”,where P=(ya·xg+yBxA),M=(ya·YB-xAxg),p=(xa+xB),m=(ya+yB)and r is theradius.The solution was verified by considering specific examples through technologysuch as Autograph software and a TI-84 graphing calculator.Finally I briefly looked atvarious other solutions to the problem and also considered further research questions.  \nWord Count:234  \nTable of Contents  \n\n| Heading   | Page   |\n| --- | --- |\n| Introduction   | 3   |\n| Pre-examination of the problem   | 4   |\n| Initial approach   | 7   |\n| Analysis of specific scenarios   | 8   |\n| Forming a general solution   | 10   |\n| Solution   | 15   |\n| Verification of solution   | 16   |\n| Other possible solutions   | 18   |\n| Further investigation   | 20   |\n| Bibliography   | 21   |\n\nExtended Essay-Mathematics  \nExtended Essay-MathematicsAlhazen's Billiard Problem  \n# Introduction:\n\nRegarded as one of the classic problems from two dimensional geometry,Alhazen's Billiard Problem has a truly rich history.The problem is believed to have beenfirst introduced by Greek astronomer Ptolemy back in 150 AD¹and then eventuallynoticed by 17ᵗcentury Arabic mathematician Abu Ali al Hassan ibn Alhaitham(whosename was later Latinized into Alhazen)².  \nAlhazen made reference to this problem in one of his published works entitledOptics and presented it in the form,\"Find the given point on a spherical mirror at which aray of light coming from a given point must strike in order to be reflected”3.Nowadays,this problem is often referred to as the “Billiard Problem”because it involves locating thepoint on the edge of a circular billiard table at which a cue ball at a given point must beaimed in order to carom (bounce)once off the edge and strike another ball at a secondgiven point.  \nThe focus question of this extended essay willbe:  \nOn a circular billiards table there are two balls;at what point along thecircumference must one be aimed at in order for it to strike the other after reboundingoff the edge?  \nHeinrich Dörrie also described the problem as \"find in a given circle an isosceles  \n⁴Eric W Weistein,“Alhazen'sBillard Problem”.Mathworld.Dated 1999.Viewed February 252006.\u003Chttp://mathworld.wolfram.com/AlhazensBilliardProblem.html>  \ntriangle whose legs pass through two given points inside the circle”.⁵  My primaryreason for choosing to investigate this focus question is that the I.B Higher LevelMathematics Programme at our school is at times limited with regards to the study ofgeometry and trigonometry.Investigating this problem gave me an opportunity to fillthis void.That being said,the problem was in itself also very appealing to me as Ipersonally enjoy playing billiards or pool and was eager to find out about themathematics of the game.  \nThe problem appeared in the ","cbCaigNBiRKypqKk","https://ap.wps.com/l/cbCaigNBiRKypqKk","pdf",7794236,23,"English","# Introduction\n## Pre-examination of the problem\n## Initial approach\n## Analysis of specific scenarios\n## Forming a general solution\n## Solution\n## Verification of solution\n## Other possible solutions\n## Further investigation\n## Bibliography","[{\"question\":\"What is the focus question of the extended essay?\",\"answer\":\"It asks: on a circular billiards table with two balls, at what point along the circumference must one aim so the ball strikes the other after rebounding off the edge.\"},{\"question\":\"What methods are used to solve the problem?\",\"answer\":\"The essay uses coordinate geometry and trigonometric principles, first trying an initial approach with chord measurements, then analyzing special cases and forming a general solution.\"},{\"question\":\"How is the final solution verified?\",\"answer\":\"The solution is checked through specific examples using technology such as Autograph software and a TI-84 graphing calculator.\"}]","Alhazen's Billiard Problem - Extended Essay - Mathematics | PDF",1790054105,58]