ISM_T11_C10_C

ISM_T11_C10_C - Section 10.2 Classifying Conic Sections by...

This preview shows pages 1–4. Sign up to view the full content.

Section 10.2 Classifying Conic Sections by Eccentricity 633 17. e take c 4 and a 5; c a b œÊ œ œ œ± 4 5 ### 16 25 b b 9 b 3; therefore Êœ ±Êœ Ê œ ## 1 x 59 y # # # ²œ 18. The eccentricity e for Pluto is 0.25 e 0.25 Êœœ œ c a4 " take c 1 and a 4; c a 1 16 b œ œ ± Ê œ ± # b 15 b 15 ; therefore, 1 is a ÊœÊ œ ² 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 œœ center is the point C( 4) and the ellipse is given by 1 ; cab325 (x 1) (y 4) 49 ±± ##### ² œ œ±œ±œ c 5 ; therefore the foci are F 1 4 5 , the ß„ ÈÈ Š‹ eccentricity is e , and the directrices are c a3 5 È y4 4 4 . œ„œ„ œ„ e5 95 È 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 ± ²œ ±Ê± ²œ ± Ê ±²² È kk # # # 24 39 x 18x 81 x y 20 5x 9y 180 or 1. œ± ² Ê² œ Ê ² œ ² œ 45 x 9 9 36 20 y ab # # # # # 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 ± ² 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. œ ± Ê ² ± œ Next, 4x y 4y 4 4 4x (y 24) 4 x 1 a 2 and b 1 (now using the # # # ± ²±²œÊ ²± œÊ ² œÊœ œ (y 2) 4 # standard symbols) c a b 4 1 3 c 3 e . Ê œ±œ±œÊœ Êœœ # 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 mm 1mm "# ± ² 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 #" # " # ³³ )) have . Then tan tan ( ) !) ) ! ) ) œ ± , since m tan and and œ tan tan m m 1t a n t a n ± " # " # ²² "" ) mt a n . œ )

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
634 Chapter 10 Conic Sections and Polar Coordinates Now we prove the reflective property of ellipses (see the accompanying figure): If 1, then x ab y # ## # ±œ b x a y a b and y a x y . w ± ± œ ² Ê œ bb x a aa x È È Let P(x y ) be any point on the ellipse !! ß y (x ) . Let F (c 0) and F ( c 0) Êœ œ ß ² ß w !" # ±± ± bx b x ay #! # ! # # É be the foci. Then m and m . Let and PF PF yy xc "# œœ ±² ! be the angles between the tangent line and PF and PF , " respectively. Then tan ! œ œ Œ± Š‹ ± bx y bxy ay( x c ) # # ! ! # # 1 ±² ± ± ² ± ± bxc ay bxc ayx ayc bxy ayc a b xy # # ! ! ! # # ! ! # ayc cxy cy b ! ! # œ . Similarly, tan . Since tan tan , and and are both less than 90°, we have . "! " ! " ! " œ b cy # ! 23. x y 1 c a b 1 1 2 e ²œÊœ ±œ ±œ Êœ ÈÈ È c a 2 ; asymptotes are y x; F 2 ; œ „ß ! È 2 1 directrices are x 0 œ„œ„ a e 2 " È 24. 9x 16y 144 1 c a b ²œÊ ² œ Ê œ± x 16 9 y # # È 16 9 5 e ; asymptotes are œ Ê œ œ È c5 a4 y x; F 5 ; directrices are x 0 œ„ „ ß! œ „ 3a 4e " 6 5 25. y x 8 1 c a b ²œÊ ²œÊœ ± y 88 x # # È 8 8 4 e 2 ; asymptotes are œ Ê œ È È c4 a 8 È y x; F 0 4 ; directrices are y 0 ß„ a e 2 È È 8 2
Section 10.2 Classifying Conic Sections by Eccentricity 635 26. y x 4 1 c a b ## ±œÊ ±œÊœ ² y 44 x # # È 4 4 2 2 e 2 ; asymptotes œ² œ Ê œ œ œ È ÈÈ c a2 22 È are y x; F 0 2 2 ; directrices are y 0

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 10/13/2011 for the course MATHEMATIC 103 taught by Professor Thommas during the Spring '11 term at LCC Intl University.

Page1 / 11

ISM_T11_C10_C - Section 10.2 Classifying Conic Sections by...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online