{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 259 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 260 "Times " 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 261 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" 258 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 261 263 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "Magenta Emphasis" -1 270 "Times" 1 12 190 0 190 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 276 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 280 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 281 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 282 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal " -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 128 0 0 1 1 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 "The Ellipse " }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: 23.3.2007" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 43 "Th e geometrical description of an ellipse " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 17 "Given two points " }{XPPEDIT 18 0 "F[1]" "6#&%\"FG6#\"\"\"" }{TEXT -1 5 " and " } {XPPEDIT 18 0 "F[2]" "6#&%\"FG6#\"\"#" }{TEXT -1 16 " in a plane, an \+ " }{TEXT 259 7 "ellipse" }{TEXT -1 36 " having each of the two points \+ as a " }{TEXT 259 5 "focus" }{TEXT -1 89 " is the collection of points P in the plane which are such that the sum of the distance F" } {XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 19 "P, from the focus F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 25 " to P, and t he distance F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 19 "P, from the focus F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 30 " to P, is constant, that is, F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT -1 5 "P + F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 15 "P is constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 70 "The picture shows an ellipse in the coordinate \+ plane with one focus at" }{XPPEDIT 18 0 "``(-2,0)" "6#-%!G6$,$\"\"#!\" \"\"\"!" }{TEXT -1 34 ", and the other focus at the point" }{XPPEDIT 18 0 "``(2,0);" "6#-%!G6$\"\"#\"\"!" }{TEXT -1 16 ", and such that " } }{PARA 256 "" 0 "" {TEXT -1 2 " F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\" \"\"" }{TEXT -1 5 "P + F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" } {TEXT -1 7 "P = 6. 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#Fgo7$$\"3')y'oMtget)Fjo$!3KfJ9k[;R@Fgo7$$\"3Z*f&\\<5EK7Fgo$!3Q)[#HLos Q?Fgo7$$\"3gdfNzeAy:Fgo$!3P&z,0Fgo7$$\"3=MN%od`Z*=Fgo$!32Z$*p;#RO t\"Fgo7$$\"3usn?(*=qd@Fgo$!33)G?:O[Nb\"Fgo7$$\"3&QL^@\"zE@CFgo$!3ms,2M -C?8Fgo7$$\"3nNq$[wo*=EFgo$!3/^&zu?<14\"Fgo7$$\"3#*H4M%o[ny#Fgo$!3)oxO ^\"[$*z#)Fjo7$$\"3*4XZ&*\\(>.HFgo$!3m.P!R8RWj&Fjo7$$\"39sa$GiE\"yHFgo$ !3GC&*\\u?G&p#FjoFeamFgam-F(6&7SFfa^q7$$\"3Oe^(ooo!zHFgo$!3)>q]v]4oj#F jo7$$\"3o.&e%f:()zHFgo$!3K&QPW%z\"fe#Fjo7$$\"3m!*e?ydv!)HFgo$!3VNPw[)o 'GDFjo7$$\"3M)z#Gjei\")HFgo$!3!z.DvIB5Z#Fjo7$$\"3ir15*)=Z#)HFgo$!3J]$G HBNOT#Fjo7$$\"37J\"[=]QK)HFgo$!3/*=%HfZTgBFjo7$$\"3[D')*4F9S)HFgo$!3yg 1C#z$H0BFjo7$$\"3CfY(*)G(z%)HFgo$!3C(Qsers#[AFjo7$$\"33&\\frFeb)HFgo$! 3!4)y;/)>9>#Fjo7$$\"3!y]b)42K')HFgo$!3cekYsY#H8#Fjo7$$\"3]e+wt^(p)HFgo $!3Q&Ri4\"**Q\"3#Fjo7$$\"3e9Dc!z#p()HFgo$!3n\"y_@kgL-#Fjo7$$\"3G(Hjj$H R))HFgo$!3')o`os#z]'>Fjo7$$\"3w4#ROH[!*)HFgo$!3oFjo7$$\"39QN5 _pi*)HFgo$!3+m)RXVwy&=Fjo7$$\"3Sj]hEYH!*HFgo$!3mc`^u)*=(z\"Fjo7$$\"370 &\\j)G%3*HFgo$!3u,BZ`*pdu\"Fjo7$$\"3#e[zzhg9*HFgo$!3!)p`d\"pWfo\"Fjo7$ $\"37&yuLW*)>*HFgo$!3I!zD`0zHj\"Fjo7$$\"3;N9hG-b#*HFgo$!3%\\[8)pn&[d\" Fjo7$$\"39\\\")GQ`1$*HFgo$!3XFJ7i1]>:Fjo7$$\"3E)>/D:$e$*HFgo$!3ou79,Ct h9Fjo7$$\"3[8oY!*4/%*HFgo$!3:<,RjWn39Fjo7$$\"33#G7X&e^%*HFgo$!3Q5XS!RN 9N\"Fjo7$$\"3%p!=1\\#))\\*HFgo$!3)*H%>Knq>H\"Fjo7$$\"3;@PQl@Q&*HFgo$!3 *fjQC)*)>S7Fjo7$$\"3K.3r6&*y&*HFgo$!3m#)*3[lwU=\"Fjo7$$\"3c,mE/1>'*HFg o$!3]3b]?c\\E6Fjo7$$\"3oal<$ejl*HFgo$!3!HUL+&3'*p5Fjo7$$\"3o%>@D=1p*HF go$!3o+8*G-a_,\"Fjo7$$\"3Ch\\S=bE(*HFgo$!3MqLmvH/X&*F^cl7$$\"3o\"zs4\\ pv*HFgo$!3k<*eUL>\"***)F^cl7$$\"3W>AK*Guy*HFgo$!3O%yV$*3qhT)F^cl7$$\"3 =d=ceG8)*HFgo$!3WNpJZ=)y)yF^cl7$$\"3+*\\*oojR)*HFgo$!3t!3Y$=HF5tF^cl7$ $\"3a'GH4-E')*HFgo$!3?mR3oKumnF^cl7$$\"3O,\")y:r%))*HFgo$!3!3K`QWb&)>' F^cl7$$\"3G!G^7aW!**HFgo$!3A`9uQ#4Ik&F^cl7$$\"3O+Pfy8B**HFgo$!3I\"yrsb m81&F^cl7$$\"3!H#HuH@R**HFgo$!3)[TSg3V6]%F^cl7$$\"3A]#o0/P&**HFgo$!39B OIlW?GRF^cl7$$\"3kq\"*)fHh'**HFgo$!3y:NSwX)*fLF^cl7$$\"33O\"[=Re(**HFg o$!3$3*>;!HIy$GF^cl7$$\"3wbct`&\\)**HFgo$!3I;#p))oj$RAF^cl7$$\"3Yjxl#) G\"***HFgo$!3muaz1V2/ " 0 "" {MPLTEXT 1 0 1 ";" } }}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 273 1 "c" }{TEXT -1 81 " be a positive real number. We shall derive the equation of the ellipse w ith foci" }{XPPEDIT 18 0 "``(-c,0)" "6#-%!G6$,$%\"cG!\"\"\"\"!" } {TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(c,0)" "6#-%!G6$%\"cG\"\"!" } {TEXT -1 90 ", and such that the sum of the distances from any point o n the ellipse to the two foci is " }{XPPEDIT 18 0 "2*a" "6#*&\"\"#\"\" \"%\"aGF%" }{TEXT -1 25 ". Note that we must have " }{XPPEDIT 18 0 "a> c" "6#2%\"cG%\"aG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{GLPLOT2D 385 378 378 {PLOTDATA 2 "67-%'CURVESG6%7S7$$\"\"$\"\"!$F* F*7$$\"3UF$)=7(3>(H!#<$\"3+`RrB[&G0$!#=7$$\"3(Q$pkH19-HF/$\"3;ey.>jgkc F27$$\"3i5$eO%fduFF/$\"3d8$4/@8S])F27$$\"3%z0RcF@ff#F/$\"3*fi&*\\R637 \"F/7$$\"3s;D')4cFrBF/$\"3?T3r&fA(p8F/7$$\"3v/=f]AxC@F/$\"31]HdOF /7$$\"3IZ7S$*=/Y6F/$\"3Y(*GmpoZm?F/7$$\"3/,WmG)ehd(F2$\"3Ob)=&)))*ej@F /7$$\"30q`%yY5a.%F2$\"3yr,u,gu:AF/7$$!3+6F@K4K\"z\"!#?$\"3Lo(z$yv1OAF/ 7$$!3A%\\Cx2Zu3%F2$\"3Y]xgOg@:AF/7$$!3hn92^kaQzF2$\"3/(GQ=Vfj:#F/7$$!3 j8$>$zE$>8\"F/$\"3%RU-+r#zq?F/7$$!3B\\n3#e%p7:F/$\"3q0o8&)y*4$>F/7$$!3 j=o!H%o%=\"=F/$\"31h&=eh(>#y\"F/7$$!3_t#HS!\\zE@F/$\"3%3#oKvx/x:F/7$$! 3Wudmh7ErBF/$\"3KA`W&RO(p8F/7$$!3M%3V>-#z(f#F/$\"307P@&z+%=6F/7$$!3-O4 %>)3GpFF/$\"3?f9v^\\O*f)F27$$!31s6#G0(3**GF/$\"3!oDQDi<3v&F27$$!3'H4De +4@(HF/$\"3?Op!*)y-?/$F27$$!3Cy\\ysb)***HF/$\"3ny\"3xrQZ$pFbo7$$!3m:ph 09`sHF/$!3&*fVP3m)*=IF27$$!3glgA!HF/$!3K4vA))pNicF27$$!3kX*4Rmy*yF F/$!3F:Dw6zwB%)F27$$!3P#[H'ReD-EF/$!3LF/7$$!3K!=;XWM=8\"F/$!3%pa8,aA32#F /7$$!3rMu(Qu'*f#yF2$!3#o$oqJQke@F/7$$!3aMOsD.quQF2$!3\"))G)G<\"Rt@#F/7 $$!3iQO5)G]@<*Fbo$!3!oS[g_dgB#F/7$$\"3%3p'*3wpJ'QF2$!3h*)pf()3XF/7$$\"3wuk%)>^!G$=F/$!3%QNQl#>Dq/j[D,Bg#F/$!3;\\&3KlhD6\"F/7$$\"3u\\Z'fG+vw#F/$!3*pA,m7_6j)F27$$ \"3qfowgmS'*GF/$!3<.`4e\"o`#eF27$$\"3mt'*ygm3sHF/$!3!R?>5T:K/$F27$F($ \"3UiT!R$3]M=!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F $6&7$7$$!\"#F*F+7$$Fb[lF*F+-Fiz6&F[[lF+$\"*++++\"!\")F+-%'SYMBOLG6$%'C IRCLEGF][l-%&STYLEG6#%&POINTG-F$6&Fe[lF[\\l-Fa\\l6$%(DIAMONDGF][lFd\\l -F$6&Fe[lF[\\l-Fa\\l6$%&CROSSGF][lFd\\l-F$6&Fe[l-Fiz6&F[[lF*F*F*-Fa\\l 6$Fc\\l\"#7Fd\\l-F$6$7%Ff[l7$$\"3#******z&['\\-)F2$\"3%******>g\"ea@F/ Fi[l-%&COLORG6&F[[lF+$\"\")F^[lF+-F$6&7#F\\^lF`\\lFd]lFd\\l-F$6&Fh^lFj \\lFd]lFd\\l-F$6&Fh^lF_]lFd]lFd\\l-%%TEXTG6&7$$!#BF^[l$F)F^[lQ)F~(-c,0 )6\"-Fb^l6&F[[l$\"\"\"Fh[lFh_lFh_l-%%FONTG6$%*HELVETICAGF][l-F^_l6&7$$ \"#%*Fh[l$\"#CF^[lQ'P(x,y)Fe_lFf_lFj_l-F^_l6&7$$\"#BF^[lFc_lQ(F~(c,0)F e_lFf_lFj_l-F^_l6&7$$!$^#Fh[l$Fb[lF^[lQ\"1Fe_lFf_l-F[`l6$F]`lFe^l-F^_l 6&7$$\"$:#Fh[lFaalQ\"2Fe_lFf_lFcal-F^_l6&7$$\"$N$Fh[l$!#8Fh[lQ\"xFe_lF f_l-F[`l6$F]`l\"\"*-F^_l6&7$$!#:Fh[l$\"#DF^[lQ\"yFe_lFf_lFcblFcbl-%+AX ESLABELSG6%Q!Fe_lFacl-F[`l6#%(DEFAULTG-%(SCALINGG6#%,CONSTRAINEDG-%*AX ESTICKSG6$F*F*-%%VIEWG6$FdclFdcl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" }}{TEXT -1 2 " " } }{PARA 0 "" 0 "" {TEXT -1 8 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((x+c)^2+(y-0)^2)+sqrt((x-c)^2+(y-0)^2) = 2*a; " "6#/,&-%%sqrtG6#,&*$,&%\"xG\"\"\"%\"cGF,\"\"#F,*$,&%\"yGF,\"\"!!\"\" F.F,F,-F&6#,&*$,&F+F,F-F3F.F,*$,&F1F,F2F3F.F,F,*&F.F,%\"aGF," }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqrt((x+c)^2+y^2)=2*a-sqrt((x-c)^2 +y^2)" "6#/-%%sqrtG6#,&*$,&%\"xG\"\"\"%\"cGF+\"\"#F+*$%\"yGF-F+,&*&F-F +%\"aGF+F+-F%6#,&*$,&F*F+F,!\"\"F-F+*$F/F-F+F8" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 44 "Squaring both sides of this equation give s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+c)^2+y^2 = \+ 4*a^2-4*a*sqrt((x-c)^2+y^2)+(x-c)^2+y^2;" "6#/,&*$,&%\"xG\"\"\"%\"cGF( \"\"#F(*$%\"yGF*F(,**&\"\"%F(*$%\"aGF*F(F(*(F/F(F1F(-%%sqrtG6#,&*$,&F' F(F)!\"\"F*F(*$F,F*F(F(F9*$,&F'F(F)F9F*F(*$F,F*F(" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 5 "Then " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2+2*c*x+c^2+y^2 = 4*a^2-4*a*sqrt((x-c)^2+y^2)+x^2-2* c*x+c^2+y^2;" "6#/,**$%\"xG\"\"#\"\"\"*(F'F(%\"cGF(F&F(F(*$F*F'F(*$%\" yGF'F(,.*&\"\"%F(*$%\"aGF'F(F(*(F0F(F2F(-%%sqrtG6#,&*$,&F&F(F*!\"\"F'F (*$F-F'F(F(F:*$F&F'F(*(F'F(F*F(F&F(F:*$F*F'F(*$F-F'F(" }{TEXT -1 2 ". \+ " }}{PARA 0 "" 0 "" {TEXT -1 34 "The last equation simplifies to: " } }{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*sqrt((x-c)^2+y^2 ) = 4*a^2-4*c*x;" "6#/*(\"\"%\"\"\"%\"aGF&-%%sqrtG6#,&*$,&%\"xGF&%\"cG !\"\"\"\"#F&*$%\"yGF1F&F&,&*&F%F&*$F'F1F&F&*(F%F&F/F&F.F&F0" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a*sqrt((x-c)^2+y^2)=a^2-c*x" "6#/*&%\"a G\"\"\"-%%sqrtG6#,&*$,&%\"xGF&%\"cG!\"\"\"\"#F&*$%\"yGF0F&F&,&*$F%F0F& *&F.F&F-F&F/" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 44 "Squaring both sides of this equation gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*((x-c)^2+y^2)=(a^2-c*x)^2" "6#/*&%\"aG\"\"#,&*$ ,&%\"xG\"\"\"%\"cG!\"\"F&F+*$%\"yGF&F+F+*$,&*$F%F&F+*&F,F+F*F+F-F&" } {TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*(x^2-2*c*x+c^2+y^2)=(a^4-2*a ^2*c*x+c^2*x^2)" "6#/*&%\"aG\"\"#,**$%\"xGF&\"\"\"*(F&F*%\"cGF*F)F*!\" \"*$F,F&F**$%\"yGF&F*F*,(*$F%\"\"%F***F&F**$F%F&F*F,F*F)F*F-*&F,F&F)F& F*" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*x^2-2*a^2*c*x+a^2*c^2+a^2*y^ 2=a^4-2*a^2*c*x+c^2*x^2" "6#/,**&%\"aG\"\"#%\"xGF'\"\"\"**F'F)*$F&F'F) %\"cGF)F(F)!\"\"*&F&F'F,F'F)*&F&F'%\"yGF'F),(*$F&\"\"%F)**F'F)*$F&F'F) F,F)F(F)F-*&F,F'F(F'F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 13 "which gives, " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 " a^2*x^2+a^2*c^2+a^2*y^2 = a^4+c^2*x^2" "6#/,(*&%\"aG\"\"#%\"xGF'\"\"\" *&F&F'%\"cGF'F)*&F&F'%\"yGF'F),&*$F&\"\"%F)*&F+F'F(F'F)" }{TEXT -1 2 " , " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "a^2*x^2-c^2*x^2+a^2*y^2 = a^4-a^2*c^2" "6#/,(*&%\" aG\"\"#%\"xGF'\"\"\"*&%\"cGF'F(F'!\"\"*&F&F'%\"yGF'F),&*$F&\"\"%F)*&F& F'F+F'F," }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 9 "that is, " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(a^2-c^2)*x^2+a^2*y^2 = a^2*(a^2-c^2);" "6#/,&*&,&*$%\"aG\"\"#\"\"\"*$%\"cGF)!\"\"F**$%\"xG F)F*F**&F(F)%\"yGF)F**&F(F),&*$F(F)F**$F,F)F-F*" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 42 "This equation can be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/(a^2)+y^2/(a ^2-c^2) = 1;" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yGF',&*$F)F'F +*$%\"cGF'F*F*F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 25 "It is convenient to let " }{XPPEDIT 18 0 "b=sqrt(a^2-c^2)" "6#/%\"bG-%% sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"cGF+!\"\"" }{TEXT -1 11 ", so that " }{XPPEDIT 18 0 "b^2=a^2-c^2" "6#/*$%\"bG\"\"#,&*$%\"aGF&\"\"\"*$%\"cGF &!\"\"" }{TEXT -1 44 ", and write the equation of the ellipse as: " }} {PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/(a^2)+y^2/(b^2) = 1;" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"yGF'*$%\"bGF'F*F+F+" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Since " }{XPPEDIT 18 0 " b = sqrt(a^2-c^2)" "6#/%\"bG-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"cGF+!\" \"" }{TEXT -1 19 ", it is clear that " }{XPPEDIT 18 0 "b(H!#<$\"3+`RrB[&G0$!#=7$$\"3(Q$pkH19-HF/$\"3;ey.>jgkc F27$$\"3i5$eO%fduFF/$\"3d8$4/@8S])F27$$\"3%z0RcF@ff#F/$\"3*fi&*\\R637 \"F/7$$\"3s;D')4cFrBF/$\"3?T3r&fA(p8F/7$$\"3v/=f]AxC@F/$\"31]HdOF /7$$\"3IZ7S$*=/Y6F/$\"3Y(*GmpoZm?F/7$$\"3/,WmG)ehd(F2$\"3Ob)=&)))*ej@F /7$$\"30q`%yY5a.%F2$\"3yr,u,gu:AF/7$$!3+6F@K4K\"z\"!#?$\"3Lo(z$yv1OAF/ 7$$!3A%\\Cx2Zu3%F2$\"3Y]xgOg@:AF/7$$!3hn92^kaQzF2$\"3/(GQ=Vfj:#F/7$$!3 j8$>$zE$>8\"F/$\"3%RU-+r#zq?F/7$$!3B\\n3#e%p7:F/$\"3q0o8&)y*4$>F/7$$!3 j=o!H%o%=\"=F/$\"31h&=eh(>#y\"F/7$$!3_t#HS!\\zE@F/$\"3%3#oKvx/x:F/7$$! 3Wudmh7ErBF/$\"3KA`W&RO(p8F/7$$!3M%3V>-#z(f#F/$\"307P@&z+%=6F/7$$!3-O4 %>)3GpFF/$\"3?f9v^\\O*f)F27$$!31s6#G0(3**GF/$\"3!oDQDi<3v&F27$$!3'H4De +4@(HF/$\"3?Op!*)y-?/$F27$$!3Cy\\ysb)***HF/$\"3ny\"3xrQZ$pFbo7$$!3m:ph 09`sHF/$!3&*fVP3m)*=IF27$$!3glgA!HF/$!3K4vA))pNicF27$$!3kX*4Rmy*yF F/$!3F:Dw6zwB%)F27$$!3P#[H'ReD-EF/$!3LF/7$$!3K!=;XWM=8\"F/$!3%pa8,aA32#F /7$$!3rMu(Qu'*f#yF2$!3#o$oqJQke@F/7$$!3aMOsD.quQF2$!3\"))G)G<\"Rt@#F/7 $$!3iQO5)G]@<*Fbo$!3!oS[g_dgB#F/7$$\"3%3p'*3wpJ'QF2$!3h*)pf()3XF/7$$\"3wuk%)>^!G$=F/$!3%QNQl#>Dq/j[D,Bg#F/$!3;\\&3KlhD6\"F/7$$\"3u\\Z'fG+vw#F/$!3*pA,m7_6j)F27$$ \"3qfowgmS'*GF/$!3<.`4e\"o`#eF27$$\"3mt'*ygm3sHF/$!3!R?>5T:K/$F27$F($ \"3UiT!R$3]M=!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F $6&7$7$$!\"#F*F+7$$Fb[lF*F+-Fiz6&F[[lF+$\"*++++\"!\")F+-%'SYMBOLG6$%'C IRCLEGF][l-%&STYLEG6#%&POINTG-F$6&Fe[lF[\\l-Fa\\l6$%(DIAMONDGF][lFd\\l -F$6&Fe[lF[\\l-Fa\\l6$%&CROSSGF][lFd\\l-F$6&Fe[l-Fiz6&F[[lF*F*F*-Fa\\l 6$Fc\\l\"#7Fd\\l-F$6$7%Ff[l7$$!3]+++C95`h!#F$\"3#)*****p(z1OAF/Fi[l-%& COLORG6&F[[lF+$\"\")F^[lF+-F$6&7#F\\^lF`\\lFd]lFd\\l-F$6&Fi^lFj\\lFd]l Fd\\l-F$6&Fi^lF_]lFd]lFd\\l-%%TEXTG6&7$$!$B#Fh[l$\"#DFh[lQ\"F6\"-Fc^l6 &F[[l$\"\"\"Fh[lFj_lFj_l-%%FONTG6$%*HELVETICAGF][l-F__l6&7$$\"$:#Fh[lF d_lFf_lFh_lF\\`l-F__l6&7$$!$8#Fh[l$\"#:Fh[lQ\"1Fg_lFh_l-F]`l6$F_`lFf^l -F__l6&7$$\"$D#Fh[lFj`lQ\"2Fg_lFh_lF]al-F__l6&7$$\"$N$Fh[l$!#8Fh[lQ\"x Fg_lFh_l-F]`l6$F_`l\"\"*-F__l6&7$$!#:Fh[l$\"#FF^[lQ\"yFg_lFh_lF]bl-F__ l6&7$$F[`lF*FjalQ\"cFg_lFh_l-F]`l6%%&TIMESG%'ITALICGF][l-F__l6&7$$F^[l F*FjalF\\clFh_lF]cl-F__l6&7$$!$E\"Fh[l$Fh]lF^[lQ\"aFg_lFh_lF]cl-F__l6& 7$$\"$E\"Fh[lFjclF[dlFh_lF]cl-F__l6&7$$F2Fh[l$\"##*Fh[lQ\"bFg_lFh_lF]c l-%+AXESLABELSG6%Q!Fg_lF[el-F]`l6#%(DEFAULTGF]bl-%*AXESTICKSG6$F*F*-%( SCALINGG6#%,CONSTRAINEDG-%%VIEWG6$F^elF^el" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve \+ 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17 " "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 62 "Now suppose that we have an ellipse with foci that are on the " }{TEXT 279 1 "y" }{TEXT -1 19 " axis at the points" }{XPPEDIT 18 0 "``(0,c)" "6#-%!G6$\"\"!%\"cG" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(0,-c)" "6#-%!G6$\"\"!,$%\"cG!\"\"" }{TEXT -1 2 ". " }}{PARA 256 " " 0 "" {TEXT -1 1 " " }{GLPLOT2D 360 421 421 {PLOTDATA 2 "67-%'CURVESG 6%7S7$$\"3#)*****p(z1OA!#<$\"\"!F,7$$\"3d4sHn*H^@#F*$\"3#H*HnZX$e4%!#= 7$$\"3Iv8!*3z7j@F*$\"3]uZyyo')*f(F27$$\"3SM'*y0o/o?F*$\"33+pX(4L49\"F* 7$$\"3QiJUya)[$>F*$\"3Rag$=GEP]\"F*7$$\"311\\G&[Wuw\"F*$\"3#e)o+,dnP=F *7$$\"3_%=A$H4U&)F2$\"3a4!=LlpC x#F*7$$\"3q(49=QNpk&F2$\"3$eTW.WgF!HF*7$$\"3Y1Kttt\"y+$F2$\"3WzI74_tsH F*7$$!3!o(o,\"zr^L\"!#?$\"3'*Q[>l%*****HF*7$$!3]BtssJgYIF2$\"3wSeG'=C? (HF*7$$!3RNEsnJ/_02f#F*7$$!3%*G3b,4Z]8F*$\"3#)[%>:,p5R#F*7$ $!3S:A=E$>_e\"F*$\"3)yE5\"=;$e6#F*7$$!3M&[q2zLuw\"F*$\"3t7NH:UpP=F*7$$ !3#R+\"R])zi$>F*$\"3;))z%G7#\\+:F*7$$!3#[\")3)*4+T1#F*$\"33+Y)f(es`6F* 7$$!33Vi62>&3;#F*$\"30\"3W)z9`:xF27$$!3u)fftDz_@#F*$\"3YHh::]F\"3%F27$ $!3II))eBs0OAF*$\"3pumZ4$GRI*Fbo7$$!3R%fvb4%f:AF*$!37ysfobR]SF27$$!3+@ F()*z'=j@F*$!3!ogE[3\\of(F27$$!3?eF*$!3/eg$fEPF\\\"F*7$$!3izIkoZzwO%)F2$!3e\"fS?)**HyFF*7$$!3W*>I!pN:LeF2$ !3%Ge\"z3Y7'*GF*7$$!3OHSF16.))GF2$!3'z^?zfs[(HF*7$$!3Bb*>)zr^OoFbo$!3; #3x&yf)***HF*7$$\"3gy_oInVzGF2$!3yBa_`D-vHF*7$$\"3)*[&RI8YKr&F2$!39X0% )))QU+HF*7$$\"3'HGuh$f5x&)F2$!3/67ldJ_qFF*7$$\"36_?x=6F*$!3#*3o;$G/ vf#F*7$$\"33$yc)yA4m8F*$!35'oav=U]P#F*7$$\"37$RG`#=X(e\"F*$!3Hrl<=m\"G 6#F*7$$\"3su)y=fLlw\"F*$!3w\"z=bc^#R=F*7$$\"3-@(oT$3kR>F*$!3#e%Rx\"4eE \\\"F*7$$\"3_R!G>#Hxi?F*$!3oe@,t0*z:\"F*7$$\"3-fn$31a)e@F*$!3(*y^->:b: yF27$$\"3@]NN-EE:AF*$!35A4W=>!H3%F27$F($\"3utR=U6ChC!#E-%'COLOURG6&%$R GBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F$6&7$7$F+$!\"#F,7$F+$Fb[lF,-Fiz 6&F[[lF+$\"*++++\"!\")F+-%'SYMBOLG6$%'CIRCLEGF][l-%&STYLEG6#%&POINTG-F $6&Fe[lF[\\l-Fa\\l6$%(DIAMONDGF][lFd\\l-F$6&Fe[lF[\\l-Fa\\l6$%&CROSSGF ][lFd\\l-F$6&Fe[l-Fiz6&F[[lF,F,F,-Fa\\l6$Fc\\l\"#7Fd\\l-F$6$7%Ff[l7$$ \"3!)*****z^j!3@F*$\"3,+++w7Y+5F*Fi[l-%&COLORG6&F[[lF+$\"\")F^[lF+-F$6 &7#F\\^lF`\\lFd]lFd\\l-F$6&Fh^lFj\\lFd]lFd\\l-F$6&Fh^lF_]lFd]lFd\\l-%% TEXTG6&7$$!#XFh[l$\"#@F^[lQ(F~(0,c)6\"-Fb^l6&F[[l$\"\"\"Fh[lFi_lFi_l-% %FONTG6$%*HELVETICAGF][l-F^_l6&7$$!#ZFh[l$!#>F^[lQ)F~(0,-c)Ff_lFg_lF[` l-F^_l6&7$$\"#DF^[l$Fj_lF,Q'P(x,y)Ff_lFg_lF[`l-F^_l6&7$$!#hFh[lFj[lQ\" 2Ff_lFg_l-F\\`l6$F^`lFe^l-F^_l6&7$$!#pFh[lFg[lQ\"1Ff_lFg_lFdal-F^_l6&7 $Fj`l$!#8Fh[lQ\"xFf_lFg_l-F\\`l6$F^`l\"\"*-F^_l6&7$$!#:Fh[l$\"$N$Fh[lQ \"yFf_lFg_lFbbl-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!Ff_lFdcl-F \\`l6#%(DEFAULTGFbbl-%*AXESTICKSG6$F,F,-%%VIEWG6$FgclFgcl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curv e 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curv e 10" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16 " }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 31 "This ellipse has t he equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/ (b^2)+y^2/(a^2) = 1;" "6#/,&*&%\"xG\"\"#*$%\"bGF'!\"\"\"\"\"*&%\"yGF'* $%\"aGF'F*F+F+" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "where \+ " }{XPPEDIT 18 0 "b = sqrt(a^2-c^2)" "6#/%\"bG-%%sqrtG6#,&*$%\"aG\"\"# \"\"\"*$%\"cGF+!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 14 " Again we have " }{XPPEDIT 18 0 "bF*$\"3Rag$=GEP]\"F*7$$\"311\\ G&[Wuw\"F*$\"3#e)o+,dnP=F*7$$\"3_%=A$H4U&)F2$\"3a4!=LlpCx#F*7$$\"3q(49=QNpk&F2$\"3$eTW.WgF!HF*7$$\"3Y 1Kttt\"y+$F2$\"3WzI74_tsHF*7$$!3!o(o,\"zr^L\"!#?$\"3'*Q[>l%*****HF*7$$ !3]BtssJgYIF2$\"3wSeG'=C?(HF*7$$!3RNEsnJ/_02f#F*7$$!3%*G3b, 4Z]8F*$\"3#)[%>:,p5R#F*7$$!3S:A=E$>_e\"F*$\"3)yE5\"=;$e6#F*7$$!3M&[q2z Luw\"F*$\"3t7NH:UpP=F*7$$!3#R+\"R])zi$>F*$\"3;))z%G7#\\+:F*7$$!3#[\")3 )*4+T1#F*$\"33+Y)f(es`6F*7$$!33Vi62>&3;#F*$\"30\"3W)z9`:xF27$$!3u)fftD z_@#F*$\"3YHh::]F\"3%F27$$!3II))eBs0OAF*$\"3pumZ4$GRI*Fbo7$$!3R%fvb4%f :AF*$!37ysfobR]SF27$$!3+@F()*z'=j@F*$!3!ogE[3\\of(F27$$!3?eF*$!3/eg$fEPF\\\"F*7$$!3izIkoZzwO%)F2$!3e\"fS?) **HyFF*7$$!3W*>I!pN:LeF2$!3%Ge\"z3Y7'*GF*7$$!3OHSF16.))GF2$!3'z^?zfs[( HF*7$$!3Bb*>)zr^OoFbo$!3;#3x&yf)***HF*7$$\"3gy_oInVzGF2$!3yBa_`D-vHF*7 $$\"3)*[&RI8YKr&F2$!39X0%)))QU+HF*7$$\"3'HGuh$f5x&)F2$!3/67ldJ_qFF*7$$ \"36_?x=6F*$!3#*3o;$G/vf#F*7$$\"33$yc)yA4m8F*$!35'oav=U]P#F*7$$\"37 $RG`#=X(e\"F*$!3Hrl<=m\"G6#F*7$$\"3su)y=fLlw\"F*$!3w\"z=bc^#R=F*7$$\"3 -@(oT$3kR>F*$!3#e%Rx\"4eE\\\"F*7$$\"3_R!G>#Hxi?F*$!3oe@,t0*z:\"F*7$$\" 3-fn$31a)e@F*$!3(*y^->:b:yF27$$\"3@]NN-EE:AF*$!35A4W=>!H3%F27$F($\"3ut R=U6ChC!#E-%'COLOURG6&%$RGBG$\"#5!\"\"F+F+-%*THICKNESSG6#\"\"#-F$6&7$7 $F+$!\"#F,7$F+$Fb[lF,-Fiz6&F[[lF+$\"*++++\"!\")F+-%'SYMBOLG6$%'CIRCLEG F][l-%&STYLEG6#%&POINTG-F$6&Fe[lF[\\l-Fa\\l6$%(DIAMONDGF][lFd\\l-F$6&F e[lF[\\l-Fa\\l6$%&CROSSGF][lFd\\l-F$6&Fe[l-Fiz6&F[[lF,F,F,-Fa\\l6$Fc\\ l\"#7Fd\\l-F$6$7%Ff[lF'Fi[l-%&COLORG6&F[[lF+$\"\")F^[lF+-F$6&7#F'F`\\l Fd]lFd\\l-F$6&Fc^lFj\\lFd]lFd\\l-F$6&Fc^lF_]lFd]lFd\\l-%%TEXTG6&7$$!#D Fh[l$\"#@F^[lQ\"F6\"-F]^l6&F[[l$\"\"\"Fh[lFd_lFd_l-%%FONTG6$%*HELVETIC AGF][l-Fi^l6&7$F\\_l$!#>F^[lF`_lFb_lFf_l-Fi^l6&7$$!#;Fh[lFj[lQ\"2Fa_lF b_l-Fg_l6$Fi_lF`^l-Fi^l6&7$Fb`lFg[lQ\"1Fa_lFb_lFe`l-Fi^l6&7$$\"#EF^[l$ !#8Fh[lQ\"xFa_lFb_l-Fg_l6$Fi_l\"\"*-Fi^l6&7$$!#:Fh[l$\"$N$Fh[lQ\"yFa_l Fb_lFcal-Fi^l6&7$$Fe_lF,F`alQ\"cFa_lFb_l-Fg_l6%%&TIMESG%'ITALICGF][l-F i^l6&7$$\"$E\"Fh[l$!#6F^[lQ\"aFa_lFb_lFcbl-Fi^l6&7$Fjbl$Fh]lF^[lF^clFb _lFcbl-Fi^l6&7$$F2Fh[l$\"##*Fh[lQ\"bFa_lFb_lFcbl-Fi^l6&7$Ffcl$!##*Fh[l FiclFb_lFcbl-%(SCALINGG6#%,CONSTRAINEDG-%+AXESLABELSG6%Q!Fa_lFfdl-Fg_l 6#%(DEFAULTGFcal-%*AXESTICKSG6$F,F,-%%VIEWG6$FidlFidl" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3 " "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 1 0" "Curve 11" "Curve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" " Curve 17" "Curve 18" "Curve 19" "Curve 20" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT 262 8 "Questio n" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Tasks" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 3 "Q1 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 5 "" 0 " " {TEXT -1 4 "Ans " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}}{PARA 0 "" 0 " " {TEXT -1 35 "___________________________________" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 35 "____________________ _______________" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 23 "Code for pictures etc. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 42 "Code for animation of ellipse construction" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 963 "aa := 3: bb := ev alf(sqrt(5)): cc := 2:\np1 := plot([aa*cos(t),bb*sin(t),t=0..2*Pi],thi ckness=2):\np2 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[circle ,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12] ):\ntt := 1.3:\np3 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt)],[cc,0]],co lor=COLOR(RGB,0,.8,0)):\np4 := plot([[[aa*cos(tt),bb*sin(tt)]]$3],styl e=point,symbol=[circle,diamond,cross],\n color=black,symbolsize=[10 $3]):\np5 := plot([[-aa,0],[aa,0]],color=blue,thickness=3):\np6 := plo t([[0,-bb],[0,bb]],color=magenta,thickness=3):\nt1 := plots[textplot]( [[3.35,-.13,`x`],[-.15,2.5,`y`]],font=[HELVETICA,9],\n color=COLOR(R GB,.01,.01,.01)):\nt2 := plots[textplot]([[-2.2,.3,`F`],[2.2,.3,`F`]], font=[TIMES,BOLD,11],\n color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[ textplot]([[-2.1,.2,`1`],[2.3,.2,`2`]],font=[HELVETICA,8],\n color=C OLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..6),t1,t2,t3],font=[HEL VETICA,9],\n scaling=constrained,tickmarks=[6,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1240 "aa := 3: bb := evalf(sqrt(5)): cc := 2:\nnu mframes := 53:\np1 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[ci rcle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3 ,12]):\nt1 := plots[textplot]([[3.35,-.13,`x`],[-.15,2.5,`y`]],font=[H ELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nfrms := NULL:\nnn : = numframes-1:\ndd := evalf(2*Pi/nn):\nfor i from 0 to nn do\n tt := dd*i;\n p2 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt)],[cc,0]],color=C OLOR(RGB,0,.8,0)):\n p3 := plot([[[aa*cos(tt),bb*sin(tt)]]$3],style= point,symbol=[circle,diamond,cross],\n color=black,symbolsize= [10$3]):\n if i=0 then\n p4 := plot([aa*cos(t),bb*sin(t),t=0..2 *Pi],color=black,linestyle=2,thickness=1):\n frms := frms,plots[d isplay]([p||(1..4),t1]); \n elif i=nn then\n p4 := plot([aa*c os(t),bb*sin(t),t=0..2*Pi],color=red,thickness=3):\n frms := frms ,plots[display]([p||(1..4),t1]);\n else \n p4 := plot([aa* cos(t),bb*sin(t),t=0..tt],color=red,thickness=3):\n p5 := plot([a a*cos(t),bb*sin(t),t=tt..2*Pi],color=black,linestyle=2,thickness=1):\n frms := frms,plots[display]([p||(1..5),t1]);\n end if: \nend do:\nplots[display]([frms],insequence=true,font=[HELVETICA,9],\n sca ling=constrained,tickmarks=[6,4]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 26 "General ellipse pictures. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 883 "aa := 3: bb := ev alf(sqrt(5)): cc := 2:\np1 := plot([aa*cos(t),bb*sin(t),t=0..2*Pi],thi ckness=2):\np2 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[circle ,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12] ):\ntt := 1.3:\np3 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt)],[cc,0]],co lor=COLOR(RGB,0,.8,0)):\np4 := plot([[[aa*cos(tt),bb*sin(tt)]]$3],styl e=point,symbol=[circle,diamond,cross],\n color=black,symbolsize=[10 $3]):\nt1 := plots[textplot]([[-2.3,.3,`F (-c,0)`],[.94,2.4,`P(x,y)`], [2.3,.3,`F (c,0)`]],font=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01)): \nt2 := plots[textplot]([[-2.51,.2,`1`],[2.15,.2,`2`]],font=[HELVETICA ,8],\n color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[3.35, -.13,`x`],[-.15,2.5,`y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.01, .01,.01)):\nplots[display]([p||(1..4),t1,t2,t3],font=[HELVETICA,9],\n \+ scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1030 "aa := 3: bb := eval f(sqrt(5)): cc := 2:\np1 := plot([aa*cos(t),bb*sin(t),t=0..2*Pi],thick ness=2):\np2 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[circle,d iamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]): \ntt := evalf(Pi/2):\np3 := plot([[-cc,0],[aa*cos(tt),bb*sin(tt)],[cc, 0]],color=COLOR(RGB,0,.8,0)):\np4 := plot([[[aa*cos(tt),bb*sin(tt)]]$3 ],style=point,symbol=[circle,diamond,cross],\n color=black,symbolsi ze=[10$3]):\nt1 := plots[textplot]([[-2.23,.25,`F`],[2.15,.25,`F`]],fo nt=[HELVETICA,10],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[te xtplot]([[-2.13,.15,`1`],[2.25,.15,`2`]],font=[HELVETICA,8],\n color =COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[3.35,-.13,`x`],[-.1 5,2.7,`y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt4 := plots[textplot]([[1,-.13,`c`],[-1,-.13,`c`],[-1.26,1.2,`a`],\n [ 1.26,1.2,`a`],[-.18,.92,`b`]],font=[TIMES,ITALIC,10],\n color=COLOR( RGB,.01,.01,.01)):\nplots[display]([p||(1..4),t||(1..4)],font=[HELVETI CA,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 884 "aa := 3: bb := evalf(sqrt(5)): cc := 2:\np1 := plot([bb*cos(t),aa*sin(t),t=0..2*P i],thickness=2):\np2 := plot([[[0,-cc],[0,cc]]$4],style=point,symbol=[ circle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10 $3,12]):\ntt := 0.34:\np3 := plot([[0,-cc],[bb*cos(tt),aa*sin(tt)],[0, cc]],color=COLOR(RGB,0,.8,0)):\np4 := plot([[[bb*cos(tt),aa*sin(tt)]]$ 3],style=point,symbol=[circle,diamond,cross],\n color=black,symbols ize=[10$3]):\nt1 := plots[textplot]([[-.45,2.1,`F (0,c)`],[-.47,-1.9,` F (0,-c)`],[2.5,1,`P(x,y)`]],font=[HELVETICA,10],color=COLOR(RGB,.01,. 01,.01)):\nt2 := plots[textplot]([[-.61,2,`2`],[-.69,-2,`1`]],font=[HE LVETICA,8],\n color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]( [[2.5,-.13,`x`],[-.15,3.35,`y`]],font=[HELVETICA,9],\n color=COLOR(R GB,.01,.01,.01)):\nplots[display]([p||(1..4),t1,t2,t3],font=[HELVETICA ,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1019 "aa := 3: b b := evalf(sqrt(5)): cc := 2:\np1 := plot([bb*cos(t),aa*sin(t),t=0..2* Pi],thickness=2):\np2 := plot([[[0,-cc],[0,cc]]$4],style=point,symbol= [circle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[1 0$3,12]):\ntt := 0:\np3 := plot([[0,-cc],[bb*cos(tt),aa*sin(tt)],[0,cc ]],color=COLOR(RGB,0,.8,0)):\np4 := plot([[[bb*cos(tt),aa*sin(tt)]]$3] ,style=point,symbol=[circle,diamond,cross],\n color=black,symbolsiz e=[10$3]):\nt1 := plots[textplot]([[-.25,2.1,`F`],[-.25,-1.9,`F`]],fon t=[HELVETICA,10],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[text plot]([[-.16,2,`2`],[-.16,-2,`1`]],font=[HELVETICA,8],\n color=COLOR (RGB,.01,.01,.01)):\nt3 := plots[textplot]([[2.6,-.13,`x`],[-.15,3.35, `y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt4 := pl ots[textplot]([[1,-.13,`c`],[1.26,-1.1,`a`],[1.26,1.2,`a`],\n [-.1 8,.92,`b`],[-.18,-.92,`b`]],font=[TIMES,ITALIC,10],\n color=COLOR(RG B,.01,.01,.01)):\nplots[display]([p||(1..4),t||(1..4)],font=[HELVETICA ,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }