PRECALCULUS ALGEBRA MAC 1140
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This 8 page Class Notes was uploaded by Nathen Fadel on Thursday September 17, 2015. The Class Notes belongs to MAC 1140 at Florida State University taught by Staff in Fall. Since its upload, it has received 45 views. For similar materials see /class/205607/mac-1140-florida-state-university in Calculus and Pre Calculus at Florida State University.
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Date Created: 09/17/15
Section 93 Ellipse App ca on De nMon Given two points F1 and F2 foci the distance of which is less than 2a where a is a positive number The set of all points P such that dPFl dPF2 2a is called an ellipse Note dAB denotes the distance between A and B If A xAyA and B xByB are two points dAB xA xB2 yA yB Examples of ellipses dFP dF2P 2a Min0r axis 1 Major axis p x y F1 to 0 x The line containing the foci Fl and F2 is called the major axis The midpoint ofthe line segment joining the foci is called the center ofthe ellipse The line through the center and perpendicular orthogonal to the major axis is called the minor axis The 2 points there the major axis intersects with the ellipse are called vertices denoted by V1 and V2 Extra Credit If F1 30 F2 30 and a5 use the definition of the ellipse and the formula of 2 2 distance between two points to show that the equation ofthe ellipse is ij 6 1 Two Fundamental Forms of Ellipse Pay attention to the differences and similarities 1 The major axis is the xaxis fat and the center is the origin From the graph we observe o a is HALF of the length ofthe MAJOR axis 0 The vertices are VI a0 and V2 a0 o b is HALF of the length ofthe MINOR axis c is HALF of the distance between the foci or is the distance between the focus and the center 0 For an ELLIPSE a b and c have the following relation Pythagorean Theorem 612 b2 02 NOTE agtb Then the corresponding equation for the ellipse is x2 y2 a2 b2 1 2 The major axis is the yaxis thin and the center is the origin wm m Pm me graph We ubseNe a mustme same as me ra1 me o a s HALF urme engm urme MAJOR am 0 The vemces are V Ura and V2 0a o n s HALF urme engm urme MWOR ast O I S HALF Elf the mstance between the Yum Dr S the mstance between the fucus and the cemer 0 Fur an ELLIPSE a n and haveme quWmg re anun Pythagurean Theurem b 2 NOTE agtb Then we unespunmng equater rurme ewpse s mrrerence 5 here Think N u L L H 25 36 What 5 b7 Example1 931aPTSclccc the graph of 7 g o Exerc 2 Ise 93 1bPTSelect the equation of the following graph i L o253s 1 L2 L 02564 1 at Ll 06425 1 Ll oxs25 1 Exercised 93 2aPTSelecL the equmuu 0139 Lhe ellipse with meme at 00 focus at 80 and vane m 7100 2 1 o 12 k 1 7 i L 054wa 1 o E 1 o 010 1 Exemlsed 932bPTFind the foci of the ellipse given by g 13 o 1153 o 10 0 My 0 0 None of these 0 1 i6 Translated Forms 0 We reprace x by xrh hgtEI than the graph ufme eguanph rs sm ed hght by h V We reprace x by gtlth WEI menthe graph unhe eguauph rs sm ed heft by h o rrhharr hwe repracey by wk my menthe graph unhe eguauph rs sm ed guwn by h n 3a 390 1h 1m x Table 3 Elli es with C liar at l1 l2 and Mujur Axis Parallel to l Coordinate Center Major Axis Foci Vertices Equation 1 Iv2 r 02 T T k ck h ak agtbandblal c39 X h1 r k T T Mr r hk a agtbandb1a1cl Ivv k Parallel to x axis h c k h a k I hi k Parallel to yaxis hk hk a I Exercise 5 9310PTSelect the equation of the following graph A 2 o 39139 2quot a i i 1 O 4 16 m4 2 72 2 O 16 y4 1 72 2 2 O FTL WTGL 1 4 2 2 2 O 11 6L Elf 4L 1 92 y4f 1 O 4 16 2 2 74 2 O km4 kyTGL 1 Exercise 6 933aPTFind the vertices of the ellipse given by 1 o 5 i 6 2 O 5 2 l 3 Exercise 7 933bPT Find the faci of the ellipse given by J 1 O 5 2 j 6 O 5 t 3 2 O 5 l 6 2 O 5 2 j 3