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ISM_T11_C10_C - Section 10.2 Classifying Conic Sections by...

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Section 10.2 Classifying Conic Sections by Eccentricity 633 17. e take c 4 and a 5; c a b œ Ê œ œ œ 4 5 # # # 16 25 b b 9 b 3; therefore Ê œ Ê œ Ê œ # # 1 x 5 9 y # œ 18. The eccentricity e for Pluto is 0.25 e 0.25 Ê œ œ œ c a 4 " take c 1 and a 4; c a b 1 16 b Ê œ œ œ Ê œ # # # # b 15 b 15 ; therefore, 1 is a Ê œ Ê œ œ # È x 16 15 y model of Pluto's orbit. 19. One axis is from A( ) to B( 7) and is 6 units long; the "ß " other axis is from C( ) to D( 1 4) and is 4 units long. $ß % ß Therefore a 3, b 2 and the major axis is vertical. The œ œ center is the point C( 4) and the ellipse is given by 1; c a b 3 2 5 (x 1) (y 4) 4 9 # # # # # œ œ œ œ c 5 ; therefore the foci are F 1 4 5 , the Ê œ ß È È Š eccentricity is e , and the directrices are œ œ c a 3 5 È y 4 4 4 . œ œ œ a 3 e 5 9 5 Š È 5 3 20. Using PF e PD, we have (x 4) y x 9 (x 4) y (x 9) x 8x 16 y œ œ Ê œ Ê È k k # # # # # # # 2 4 3 9 x 18x 81 x y 20 5x 9y 180 or 1. œ Ê œ Ê œ œ 4 5 x 9 9 36 20 y a b # # # # # 21. The ellipse must pass through ( 0) c 0; the point ( 1 2) lies on the ellipse a 2b 8. The ellipse Ê œ ß Ê œ is tangent to the x-axis its center is on the y-axis, so a 0 and b 4 the equation is 4x y 4y 0. Ê œ œ Ê œ # # Next, 4x y 4y 4 4 4x (y 24) 4 x 1 a 2 and b 1 (now using the # # # # # œ Ê œ Ê œ Ê œ œ (y 2) 4 standard symbols) c a b 4 1 3 c 3 e . Ê œ œ œ Ê œ Ê œ œ # # # # È c a 3 È 22. We first prove a result which we will use: let m , and " m be two nonparallel, nonperpendicular lines. Let be # ! the acute angle between the lines. Then tan . ! œ m m 1 m m To see this result, let be the angle of inclination of the ) " line with slope m , and be the angle of inclination of the " # ) line with slope m . Assume m m . Then and we # " # " # ) ) have . Then tan tan ( ) ! ) ) ! ) ) œ œ " # " # , since m tan and and œ œ œ tan tan m m 1 tan tan 1 m m ) ) ) ) " " ) m tan . # # œ )
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634 Chapter 10 Conic Sections and Polar Coordinates Now we prove the reflective property of ellipses (see the accompanying figure): If 1, then x a b y œ b x a y a b and y a x y . # # # # # # w # # œ œ Ê œ b bx a a a x È È Let P(x y ) be any point on the ellipse ! ! ß y (x ) . Let F (c 0) and F ( c 0) Ê œ œ ß ß w ! " # bx b x a a x a y É be the foci. Then m and m . Let and PF PF y y x c x c œ œ ! be the angles between the tangent line and PF and PF , " " # respectively. Then tan ! œ œ œ œ Œ Š b x y a y x c b x y a y (x c) 1 b x b x c a y b x c b x a y a y x a y c b x y a y c a b x y b x c a b a b a b a y c c x y cy b œ . Similarly, tan . Since tan tan , and and are both less than 90°, we have . " ! " ! " ! " œ œ œ b cy 23. x y 1 c a b 1 1 2 e # # # # œ Ê œ œ œ Ê œ È È È c a 2 ; asymptotes are y x; F 2 ; œ œ œ „ ß ! È 2 1 È È Š directrices are x 0 œ œ „ a e 2 " È 24. 9x 16y 144 1 c a b # # # # œ Ê œ Ê œ x 16 9 y È 16 9 5 e ; asymptotes are œ œ Ê œ œ È c 5 a 4 y x; F 5 ; directrices are x 0 œ „ ß ! œ 3 a 4 e a b œ „ " 6 5 25. y x 8 1 c a b # # # # œ Ê œ Ê œ y 8 8 x È 8 8 4 e 2 ; asymptotes are œ œ Ê œ œ œ È È c 4 a 8 È y x; F 0 4 ; directrices are y 0 œ „ ß „ œ a b a e 2 œ „ œ „ È È 8 2
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Section 10.2 Classifying Conic Sections by Eccentricity 635 26. y x 4 1 c a b # # # # œ Ê œ Ê œ y 4 4 x È 4 4 2 2 e
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