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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 "Maple Plot" -1 13 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 }{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 }{PSTYLE "Normal " -1 257 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 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 15 "The Hyperbola " }}{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 44 "Th e geometrical description of a hyperbola " }}{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 15 " in a plane, a " }{TEXT 259 9 "hyperbola" }{TEXT -1 36 " having each of the two points \+ as a " }{TEXT 259 5 "focus" }{TEXT -1 96 " is the collection of points P in the plane which are such that the difference of the distance F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 19 "P, from the focu s F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 25 " to P, and \+ the distance F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 19 "P , from the focus F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 33 " 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 18 "P | is constant. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 71 "The picture shows a hyperbola in the coor dinate plane with one focus at" }{XPPEDIT 18 0 "``(-sqrt(13),0);" "6#- %!G6$,$-%%sqrtG6#\"#8!\"\"\"\"!" }{TEXT -1 34 ", and the other focus a t the point" }{XPPEDIT 18 0 "``(sqrt(13),0);" "6#-%!G6$-%%sqrtG6#\"#8 \"\"!" }{TEXT -1 16 ", and such that " }}{PARA 256 "" 0 "" {TEXT -1 3 "| F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" }{TEXT -1 5 "P - F" } {XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 9 "P | = 6. 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" }}{PARA 0 "" 0 "" {TEXT -1 154 "If a point P moves along one of the branches of the hyperbola, then any change of position of P is accompanied by corresponding chan ges in the distances F" }{XPPEDIT 18 0 "``[1]" "6#&%!G6#\"\"\"" } {TEXT -1 7 "P and F" }{XPPEDIT 18 0 "``[2]" "6#&%!G6#\"\"#" }{TEXT -1 196 "P of the point P from the two foci. These changes in distance mus t be equal, provided that the point stays on the same side of the line of symmetry of the hyperbola which passes through the foci. " }} {PARA 0 "" 0 "" {TEXT -1 149 "The two points where the line of symmetr y which passes through the foci of the hyperbola intersects the two br anches of the hyperbola are called the " }{TEXT 259 8 "vertices" } {TEXT -1 19 " of the hyperbola. " }}{PARA 0 "" 0 "" {TEXT -1 44 "The m id-point of the two foci is called the " }{TEXT 259 6 "centre" }{TEXT -1 73 " of the hyperbola. The centre is also the mid-point of the two \+ vertices. " }}{PARA 0 "" 0 "" {TEXT -1 138 "The line segment cut-off f rom the line of symmetry passing through the foci and centre by the tw o branches of the hyperbola is called the " }{TEXT 259 15 "transverse \+ axis" }{TEXT -1 18 " of the hyperbola." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 55 "The centre of the hyperbola shown abo ve is the origin. " }}{PARA 0 "" 0 "" {TEXT -1 30 "This hyperbola inte rsects the " }{TEXT 268 1 "x" }{TEXT -1 19 " axis at the points" } {XPPEDIT 18 0 "``(-3,0)" "6#-%!G6$,$\"\"$!\"\"\"\"!" }{TEXT -1 4 " and " }{XPPEDIT 18 0 " ``(3,0)" "6#-%!G6$\"\"$\"\"!" }{TEXT -1 54 ", and t hese points are the vertices of the hyperbola. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 28 "The equation of \+ a hyperbola " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 "" 0 "" {TEXT -1 4 "Let " }{TEXT 269 1 "c" }{TEXT -1 83 " be a posit ive real number. We shall derive the equation of the hyperbola with fo ci" }{XPPEDIT 18 0 "``(-c,0)" "6#-%!G6$,$%\"cG!\"\"\"\"!" }{TEXT -1 4 " and" }{XPPEDIT 18 0 " ``(c,0)" "6#-%!G6$%\"cG\"\"!" }{TEXT -1 99 ", \+ and such that the difference of the distances from any point on the hy perbola to the two foci is " }{XPPEDIT 18 0 "2*a" "6#*&\"\"#\"\"\"%\"a GF%" }{TEXT -1 25 ". Note that we must have " }{XPPEDIT 18 0 "a < c;" "6#2%\"aG%\"cG" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {GLPLOT2D 446 321 321 {PLOTDATA 2 "68-%'CURVESG6%7W7$$\"3+A%zhR&p_k!#< $!3>oI!H+.'3QF*7$$\"35d)z*[zL\"G'F*$!3_k`W*R#3zOF*7$$\"3GZ#3'3.$e6'F*$ !33q'\\N7))Hb$F*7$$\"3y8>0s@YwfF*$!3'Rz@uFqfW$F*7$$\"30BEY%o28%eF*$!3v >#H0)>QTLF*7$$\"3/m7umaA_bF*$!3gm.#R\")RY6$F*7$$\"3;J'ywSI7G&F*$!3UjRb lah(*GF*7$$\"3!>!=Vl4[I]F*$!3m\\:rIV-#p#F*7$$\"3d6Fr9X69[F*$!3RcRBE>/5 DF*7$$\"3TMz$=g&o0YF*$!3]*yTF*QuHBF*7$$\"3#R)e-ur51WF*$!3mmSc/$f8:#F*7 $$\"3oB3S#HYEA%F*$!3gOwF*7$$\"3#zj[gfy$\\SF*$!3&[iDK**=K\"=F*7 $$\"3%pr3`,I#4RF*$!3%RqYRVQ4n\"F*7$$\"3'e`0Lf,\\w$F*$!3340.z%*\\;:F*7$ $\"3(*3vTw,!Qj$F*$!3]loJOC*pO\"F*7$$\"3Ec&4g!>6?NF*$!3W!4Q\"[5\"F*7$$\"3E\"4$eS9[HLF*$!3<0>mvPQF'*!# =7$$\"3]5judo+dKF*$!35Y&z*=A/a%)Faq7$$\"3i%3yIg,W=$F*$!3cjC%f>V$>rFaq7 $$\"3)e(QjPzTIJF*$!3epSJOc/hfFaq7$$\"3%R#zA/v4#3$F*$!3Y!=hLFi3r%Faq7$$ \"39Lamiy`YIF*$!3$*ow,-$F*$!3U)*3[<$[MK#Faq7$ $\"3HMU(G<\\b+$F*$!3Z%Q4+DCq@\"Faq7$$\"3f9s4lG++IF*$!3MF@gYD5kF!#?7$$ 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$$!30BEY%o28%eF*F?7$$!3/m7umaA_bF*FD7$$!3;J'ywSI7G&F*FI7$$!3!>!=Vl4[I] F*FN7$$!3d6Fr9X69[F*FS7$$!3TMz$=g&o0YF*FX7$$!3#R)e-ur51WF*Fgn7$$!3oB3S #HYEA%F*F\\o7$$!3#zj[gfy$\\SF*Fao7$$!3%pr3`,I#4RF*Ffo7$$!3'e`0Lf,\\w$F *F[p7$$!3(*3vTw,!Qj$F*F`p7$$!3Ec&4g!>6?NF*Fep7$$!31(z^\"3$*HFMF*Fjp7$$ !3E\"4$eS9[HLF*F_q7$$!3]5judo+dKF*Feq7$$!3i%3yIg,W=$F*Fjq7$$!3)e(QjPzT IJF*F_r7$$!3%R#zA/v4#3$F*Fdr7$$!39Lamiy`YIF*Fir7$$!3CY_\"))>w,-$F*F^s7 $$!3HMU(G<\\b+$F*Fcs7$$!3f9s4lG++IF*Fhs7$$!3S(y5#y\\Y0IF*F^t7$$!3RD0w_ da>IF*Fct7$$!3X`<()e@bWIF*Fht7$$!3meH;45;\"3$F*F]u7$$!3+=6GmspFJF*Fbu7 $$!3'**o/MYoH=$F*Fgu7$$!3+\"pK2+ijD$F*F\\v7$$!3HSFGbQLLLF*Fav7$$!3$ecP /UDtU$F*Ffv7$$!3dhAqr!3L_$F*F[w7$$!36AEoElNSOF*F`w7$$!3'3G#*[&)4Cw$F*F ew7$$!3bW()4oRx-RF*Fjw7$$!318i96')4`SF*F_x7$$!3#)z@o.@*[A%F*Fdx7$$!3?' \\%*GKm[S%F*Fix7$$!3$G`A$)y(G/YF*F^y7$$!3St4))es4=[F*Fcy7$$!36BR&H+&GH ]F*Fhy7$$!3Atm)*f\")[*G&F*F]z7$$!3==*fuYI%RbF*Fbz7$$!3GX&4L'QsCeF*Fgz7 $$!3o,hfHESofF*F\\[l7$$!3QrmN+R'o6'F*Fa[l7$$!3)>CUd0t=G'F*Ff[l7$F\\]lF i[lF[\\lFd\\l-F$6&7$7$$!37*)RYv7b0OF*Fb\\l7$$\"37*)RYv7b0OF*Fb\\l-F\\ \\l6&F^\\lFb\\l$\"*++++\"!\")Fb\\l-%'SYMBOLG6$%'CIRCLEGF`\\l-%&STYLEG6 #%&POINTG-F$6&FjflFagl-Fggl6$%(DIAMONDGF`\\lFjgl-F$6&FjflFagl-Fggl6$%& CROSSGF`\\lFjgl-F$6&Fjfl-F\\\\l6&F^\\lFc\\lFc\\lFc\\l-Fggl6$Figl\"#7Fj gl-F$6$7%F[gl7$$\"3(********oUF\"fF*$\"3/+++u[w'R$F*F^gl-%&COLORG6&F^ \\lFb\\l$\"\")Fa\\lFb\\l-F$6&7#FbilFfglFjhlFjgl-F$6&F^jlF`hlFjhlFjgl-F $6&F^jlFehlFjhlFjgl-%%TEXTG6&7$$!#TFa\\l$!\"%Fa\\lQ)F~(-c,0)6\"-Fhil6& F^\\l$\"\"\"!\"#F_[mF_[m-%%FONTG6$%*HELVETICAGF`\\l-Fdjl6&7$$\"#_Fa\\l $\"#TFa\\lQ'P(x,y)F\\[mF][mFb[m-Fdjl6&7$F[\\mFijlQ(F~(c,0)F\\[mF][mFb[ m-Fdjl6&7$$!$`%Fa[m$!\"'Fa\\lQ\"1F\\[mF][m-Fc[m6$Fe[mF[jl-Fdjl6&7$$\"$ t$Fa[mFg\\mQ\"2F\\[mF][mFj\\m-Fdjl6&7$$\"\"(Fc\\l$Fa[mFa\\lQ\"xF\\[mF] [m-Fc[m6$Fe[m\"\"*-Fdjl6&7$Fg]m$\"#XFa\\lQ\"yF\\[mF][mFi]m-%(SCALINGG6 #%,CONSTRAINEDG-%*AXESTICKSG6$Fc\\lFc\\lFi]m-%+AXESLABELSG6%Q!F\\[mF\\ _m-Fc[m6#%(DEFAULTG-%%VIEWG6$F__mF__m" 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 " }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "We have " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(sqrt((x+c)^2+(y-0)^2)-s qrt((x-c)^2+(y-0)^2)) = 2*a;" "6#/-%$absG6#,&-%%sqrtG6#,&*$,&%\"xG\"\" \"%\"cGF/\"\"#F/*$,&%\"yGF/\"\"!!\"\"F1F/F/-F)6#,&*$,&F.F/F0F6F1F/*$,& F4F/F5F6F1F/F6*&F1F/%\"aGF/" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 8 "so that " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "sqr t((x+c)^2+(y-0)^2)-sqrt((x-c)^2+(y-0)^2) = ``;" "6#/,&-%%sqrtG6#,&*$,& %\"xG\"\"\"%\"cGF,\"\"#F,*$,&%\"yGF,\"\"!!\"\"F.F,F,-F&6#,&*$,&F+F,F-F 3F.F,*$,&F1F,F2F3F.F,F3%!G" }{TEXT 273 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "2*a" "6#*&\"\"#\"\"\"%\"aGF%" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 12 "This gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " \+ " }{XPPEDIT 18 0 "sqrt((x+c)^2+y^2) = ``;" "6#/-%%sqrtG6#,&*$,&%\"xG\" \"\"%\"cGF+\"\"#F+*$%\"yGF-F+%!G" }{TEXT 274 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "2*a+sqrt((x-c)^2+y^2);" "6#,&*&\"\"#\"\"\"%\"aGF&F&-%%s qrtG6#,&*$,&%\"xGF&%\"cG!\"\"F%F&*$%\"yGF%F&F&" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 44 "Squaring both sides of this equation give s: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "(x+c)^2+y^2 = \+ 4*a^2;" "6#/,&*$,&%\"xG\"\"\"%\"cGF(\"\"#F(*$%\"yGF*F(*&\"\"%F(*$%\"aG F*F(" }{TEXT -1 1 " " }{TEXT 275 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*sqrt((x-c)^2+y^2)+(x-c)^2+y^2" "6#,(*(\"\"%\"\"\"%\"aGF&-%%sqrtG6 #,&*$,&%\"xGF&%\"cG!\"\"\"\"#F&*$%\"yGF1F&F&F&*$,&F.F&F/F0F1F&*$F3F1F& " }{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;" "6#/ ,**$%\"xG\"\"#\"\"\"*(F'F(%\"cGF(F&F(F(*$F*F'F(*$%\"yGF'F(*&\"\"%F(*$% \"aGF'F(" }{TEXT -1 1 " " }{TEXT 276 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*sqrt((x-c)^2+y^2)+x^2-2*c*x+c^2+y^2" "6#,,*(\"\"%\"\"\"%\"aG F&-%%sqrtG6#,&*$,&%\"xGF&%\"cG!\"\"\"\"#F&*$%\"yGF1F&F&F&*$F.F1F&*(F1F &F/F&F.F&F0*$F/F1F&*$F3F1F&" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 34 "The last equation simplifies to: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 277 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "4*a*sqrt((x- c)^2+y^2) = 4*a^2-4*c*x;" "6#/*(\"\"%\"\"\"%\"aGF&-%%sqrtG6#,&*$,&%\"x GF&%\"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 278 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a*sqrt((x-c)^2+y^2)= a^2-c*x" "6#/*&%\"aG\"\"\"-%%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 "t hat 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**$%\"xGF)F*F**&F(F)%\"yGF)F**&F(F),&*$F(F )F**$F,F)F-F*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 103 "This exactly the same equation as is obt ained when deriving the equation of an ellipse. However, since " } {XPPEDIT 18 0 "aU!)p5%F*7$$\"37u!QP)[JamF*$!3-cq\\JFzfRF*7$$\"3IE-CP/,)['F*$!3 ()*zhom&F*$!3FGUyX9D,KF*7$$\"3o'Rf@*fel`F*$!3)yso5q$ plHF*7$$\"3E[\\l\\&>06&F*$!3AsKK#z7#eFF*7$$\"3qs<\"[!3Vl[F*$!3iV&o**RR Ob#F*7$$\"3lK6D9sLJYF*$!3KLsxIiB_BF*7$$\"3&zbUF/rmT%F*$!3Ww*oQ7l4;#F*7 $$\"3)zAOBh/W@%F*$!3')pk4!**)Gt>F*7$$\"3a[7w]]9^SF*$!3%y4**f$=(\\\"=F* 7$$\"31zk`!pYL)QF*$!3/QXw3`*Qk\"F*7$$\"3eP.[TMAJPF*$!3719U*)H/z9F*7$$ \"3_\\1$QgR&*f$F*$!3[9?uwX4E8F*7$$\"3'fu!*p=+A\\$F*$!3k[z%)3Ps\">\"F*7 $$\"3kVXci[BzLF*$!3bfkNnX#p.\"F*7$$\"3SF=M]nk&H$F*$!31_a0(Rn_4*!#=7$$ \"3d*>DUp-?@$F*$!3SI!4>k&f]wFfq7$$\"3#Rwlolt)\\JF*$!3k!fm\"*Gh.S'Ffq7$ $\"3'o^J`&yI%4$F*$!37q'zG47T0&Ffq7$$\"3ASDUVRW`IF*$!3=#*fRU2#>z$Ffq7$$ \"3Kjk3*GlJ-$F*$!3k**Q\"f0L-\\#Ffq7$$\"3n,u6%4PFfq7$ $\"3y(o+cOJK4$F*$\"3JlC_U+uC]Ffq7$$\"3%\\b+2*QuYJF*$\"3'*RB*Rdo:L'Ffq7 $$\"3'f.>^i_.@$F*$\"3$)p\\)\\/_(>wFfq7$$\"311W%3V.\\H$F*$\"3%zZ$G)>!H$ 3*Ffq7$$\"3!y[VHk!o$Q$F*$\"3Uw)3u<[L/\"F*7$$\"3m7WGl.B#\\$F*$\"3Q\"zvA ,j<>\"F*7$$\"3<2%*fu'QKg$F*$\"3ovSEB>bI8F*7$$\"3jnwRmU#)QPF*$\"3+:^d[$ \\v[\"F*7$$\"3eb26[FX!)QF*$\"3%ej**)o_&3k\"F*7$$\"3\"=h^n$>jVSF*$\"39X pqk,^2=F*7$$\"3.Z'H(4Bu=UF*$\"3g)y^/@/u(>F*7$$\"3%H$HjkdH>WF*$\"3E8(zs u[L;#F*7$$\"3iYgq0O))HYF*$\"3C]KqwS'4N#F*7$$\"39>39K#*yj[F*$\"3o1.bx\" \\Ab#F*7$$\"3/:aY#z3_6&F*$\"3gxB/vL2iFF*7$$\"3S#yU%[VXrgRF*7$$\"3#fO621\\F&oF*$\"3^\\14 %*\\X2TF*7$F($\"3iM'*=5*e&eUF*-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fc\\lF b\\l-%*THICKNESSG6#\"\"#-F$6%7W7$$!3iT(Hd%)Gs0(F*F+7$$!39<.-\\!4@&oF*F 07$$!37u!QP)[JamF*F57$$!3IE-CP/,)['F*F:7$$!3`O$f!fr&pK'F*F?7$$!3Kq0Gb[ B$)fF*FD7$$!35]s\\-1)>m&F*FI7$$!3o'Rf@*fel`F*FN7$$!3E[\\l\\&>06&F*FS7$ $!3qs<\"[!3Vl[F*FX7$$!3lK6D9sLJYF*Fgn7$$!3&zbUF/rmT%F*F\\o7$$!3)zAOBh/ W@%F*Fao7$$!3a[7w]]9^SF*Ffo7$$!31zk`!pYL)QF*F[p7$$!3eP.[TMAJPF*F`p7$$! 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3]BAA-tn6JF*7$F`fm$!3qAAAP'H'4LF*7$Fefm$!3[XW%p#G%f]$F*7$Fjfm$!33,++gU L'o$F*7$F_gm$!3![WW%*>$3$*QF*7$Fdgm$!3Obbb:>+ySF*7$Figm$!3ELL$e$Q;vUF* 7$F^hm$!39ML$3&H(QY%F*7$FchmFeilFjhlFghm-F$6%7'7$$FjhmFc\\l$Fg\\lFc\\l 7$$!\"$Fc\\lFbbn7$Fdbn$!\"#Fc\\l7$FabnFgbnF`bn-F\\\\l6&F^\\l$\")!\\DP \"FeglF\\cn$\")viobFegl-FhhmFf\\l-%%TEXTG6&7$$!#PFa\\l$\"\"&Fa\\lQ\"F6 \"-%&COLORG6&F^\\l$\"\"\"FhbnF^dnF^dn-%%FONTG6$%*HELVETICAGF`\\l-Fbcn6 &7$$\"#PFa\\lFgcnFicnF[dnF`dn-Fbcn6&7$Fchm$FhbnFa\\lQ\"xFjcnF[dn-Fadn6 $Fcdn\"\"*-Fbcn6&7$F\\en$\"#XFa\\lQ\"yFjcnF[dnF^en-Fbcn6&7$$!+v7b0O!\" *$FebnFa\\lQ#-cFjcnF[dnF^en-Fbcn6&7$$\"+v7b0OF\\fnF]fnQ\"cFjcnF[dnF^en -Fbcn6&7$$!$#GFhbnF]fnQ#-aFjcnF[dnF^en-Fbcn6&7$$\"$&GFhbnF]fnQ\"aFjcnF [dnF^en-Fbcn6&7$F]fnFgbnQ#-bFjcnF[dnF^en-Fbcn6&7$F]fnFbbnQ\"bFjcnF[dnF ^en-Fbcn6&7$$!#NFa\\l$\"\"%Fa\\lQ\"1FjcnF[dn-Fadn6$Fcdn\"\")-Fbcn6&7$$ \"#RFa\\lF^hnQ\"2FjcnF[dnFahn-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6 $Fc\\lFc\\lF^en-%+AXESLABELSG6%Q!FjcnFdin-Fadn6#%(DEFAULTG-%%VIEWG6$;F cilFchmFgin" 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" "Curve 21" }}{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT 259 5 "Notes" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 51 "The equation (ii) may a lso be written in the form: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT 304 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "b/a;" "6#*&%\"bG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x*sqrt(1-a^2/(x^2))" "6#*&%\"xG\"\"\"-%%sqrtG6#,&F%F%*& %\"aG\"\"#*$F$F,!\"\"F.F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "Then one can observe that, when " }{TEXT 305 1 "x" }{TEXT -1 47 " is large and positive, or large and negative, " }{XPPEDIT 18 0 "a ^2/(x^2)" "6#*&%\"aG\"\"#*$%\"xGF%!\"\"" }{TEXT -1 1 " " }{TEXT 315 1 "~" }{TEXT -1 12 " 0, giving " }{TEXT 307 1 "y" }{TEXT -1 1 " " } {TEXT 306 1 "~" }{TEXT -1 1 " " }{TEXT 308 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "b/a;" "6#*&%\"bG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " } {TEXT 309 1 "x" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 128 "The equations of the asymptotes can be o btained quickly from equation (i) by replacing the 1 on the right side by 0 to obtain: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "x^2/(a^2)-y^2/(b^2) = 0;" "6#/,&*&%\"xG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\" yGF'*$%\"bGF'F*F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "This equation is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/(b^2) = x^2/(a^2);" "6#/*&%\"yG\"\"#*$%\"bGF&! \"\"*&%\"xGF&*$%\"aGF&F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y/b =`` " " 6#/*&%\"yG\"\"\"%\"bG!\"\"%!G" }{TEXT 310 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "x/a" "6#*&%\"xG\"\"\"%\"aG!\"\"" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 3 "or " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y =``" "6#/%\"yG%!G" }{TEXT 311 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "b/a" "6#*&%\"bG\"\"\"%\"aG!\"\"" }{TEXT -1 1 " " } {TEXT 312 1 "x" }{TEXT -1 1 "." }}{PARA 256 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 17 "The relationship " }{XPPEDIT 18 0 "a^2+b^ 2=c^2" "6#/,&*$%\"aG\"\"#\"\"\"*$%\"bGF'F(*$%\"cGF'" }{TEXT -1 9 " bet ween " }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 5 " and " }{TEXT 317 1 "c" }{TEXT -1 244 " can be pictured as an application of Pythago ras' theorem in any of the four right-angled triangles into which the \+ central rectangle is disected by the asymptotes which form the diagona ls. The hypotenuse of each of this rectangles is of length " }{TEXT 316 1 "c" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 63 "Now suppose that we have a hyperbola with foci th at are on the " }{TEXT 272 1 "y" }{TEXT -1 19 " axis at the points" } {XPPEDIT 18 0 "``(0,c)" "6#-%!G6$\"\"!%\"cG" }{TEXT -1 4 " and" } {XPPEDIT 18 0 " ``(0,-c)" "6#-%!G6$\"\"!,$%\"cG!\"\"" }{TEXT -1 2 ". \+ " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 346 366 366 {PLOTDATA 2 "68-%'CURVESG6%7W7$$!3>oI!H+.'3Q!#<$\"3+A%zhR&p_kF*7$$!3_k`W*R#3zOF* $\"35d)z*[zL\"G'F*7$$!33q'\\N7))Hb$F*$\"3GZ#3'3.$e6'F*7$$!3'Rz@uFqfW$F *$\"3y8>0s@YwfF*7$$!3v>#H0)>QTLF*$\"30BEY%o28%eF*7$$!3gm.#R\")RY6$F*$ \"3/m7umaA_bF*7$$!3UjRblah(*GF*$\"3;J'ywSI7G&F*7$$!3m\\:rIV-#p#F*$\"3! >!=Vl4[I]F*7$$!3RcRBE>/5DF*$\"3d6Fr9X69[F*7$$!3]*yTF*QuHBF*$\"3TMz$=g& o0YF*7$$!3mmSc/$f8:#F*$\"3#R)e-ur51WF*7$$!3gOwF*$\"3oB3S#HYEA% F*7$$!3&[iDK**=K\"=F*$\"3#zj[gfy$\\SF*7$$!3%RqYRVQ4n\"F*$\"3%pr3`,I#4R F*7$$!3340.z%*\\;:F*$\"3'e`0Lf,\\w$F*7$$!3]loJOC*pO\"F*$\"3(*3vTw,!Qj$ F*7$$!3W!46?NF*7$$!3_')G%*>Q\"[5\"F*$\"31(z^\"3$* HFMF*7$$!3<0>mvPQF'*!#=$\"3E\"4$eS9[HLF*7$$!35Y&z*=A/a%)F_q$\"3]5judo+ dKF*7$$!3cjC%f>V$>rF_q$\"3i%3yIg,W=$F*7$$!3epSJOc/hfF_q$\"3)e(QjPzTIJF *7$$!3Y!=hLFi3r%F_q$\"3%R#zA/v4#3$F*7$$!3$*ow,-$F*7$$!3Z%Q4+DCq@\"F_q$\"3HMU(G<\\b+ $F*7$$!3MF@gYD5kF!#?$\"3f9s4lG++IF*7$$\"3iS$)yiyu27F_q$\"3S(y5#y\\Y0IF *7$$\"31Rk)p1SnG#F_q$\"3RD0w_da>IF*7$$\"3!4L#HwVffMF_q$\"3X`<()e@bWIF* 7$$\"3U(Rd[$fb$o%F_q$\"3meH;45;\"3$F*7$$\"3%H3GFqJs*eF_q$\"3+=6GmspFJF *7$$\"3q*e!H-B!34(F_q$\"3'**o/MYoH=$F*7$$\"3\"\\ZJIT%*HW)F_q$\"3+\"pK2 +ijD$F*7$$\"3?T\"['z+X'o*F_q$\"3HSFGbQLLLF*7$$\"3o\"*RQ@)\\[5\"F*$\"3$ ecP/UDtU$F*7$$\"35W%z`oY5Nsu8F*$\"36A EoElNSOF*7$$\"3*))yFd`[P^\"F*$\"3'3G#*[&)4Cw$F*7$$\"3yfwf'[F*$\"3# )z@o.@*[A%F*7$$\"3c'fa^oH-:#F*$\"3?'\\%*GKm[S%F*7$$\"3Ur6L,c^GBF*$\"3$ G`A$)y(G/YF*7$$\"3x(G*3,gV8DF*$\"3St4))es4=[F*7$$\"3mNs_**4.\"p#F*$\"3 6BR&H+&GH]F*7$$\"33h3Us@I/HF*$\"3Atm)*f\")[*G&F*7$$\"3`!GIa!y\\/JF*$\" 3==*fuYI%RbF*7$$\"31Zm3a.\\GLF*$\"3GX&4L'QsCeF*7$$\"3l[+.[kvRMF*$\"3o, hfHESofF*7$$\"3Wn$GB$)yPb$F*$\"3QrmN+R'o6'F*7$$\"3'RiofV)[zOF*$\"3)>CU d0t=G'F*7$$\"3>oI!H+.'3QF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!Fc\\lFb \\l-%*THICKNESSG6#\"\"#-F$6%7W7$F($!3+A%zhR&p_kF*7$F.$!35d)z*[zL\"G'F* 7$F3$!3GZ#3'3.$e6'F*7$F8$!3y8>0s@YwfF*7$F=$!30BEY%o28%eF*7$FB$!3/m7uma A_bF*7$FG$!3;J'ywSI7G&F*7$FL$!3!>!=Vl4[I]F*7$FQ$!3d6Fr9X69[F*7$FV$!3TM z$=g&o0YF*7$Fen$!3#R)e-ur51WF*7$Fjn$!3oB3S#HYEA%F*7$F_o$!3#zj[gfy$\\SF *7$Fdo$!3%pr3`,I#4RF*7$Fio$!3'e`0Lf,\\w$F*7$F^p$!3(*3vTw,!Qj$F*7$Fcp$! 3Ec&4g!>6?NF*7$Fhp$!31(z^\"3$*HFMF*7$F]q$!3E\"4$eS9[HLF*7$Fcq$!3]5judo +dKF*7$Fhq$!3i%3yIg,W=$F*7$F]r$!3)e(QjPzTIJF*7$Fbr$!3%R#zA/v4#3$F*7$Fg r$!39Lamiy`YIF*7$F\\s$!3CY_\"))>w,-$F*7$Fas$!3HMU(G<\\b+$F*7$Ffs$!3f9s 4lG++IF*7$F\\t$!3S(y5#y\\Y0IF*7$Fat$!3RD0w_da>IF*7$Fft$!3X`<()e@bWIF*7 $F[u$!3meH;45;\"3$F*7$F`u$!3+=6GmspFJF*7$Feu$!3'**o/MYoH=$F*7$Fju$!3+ \"pK2+ijD$F*7$F_v$!3HSFGbQLLLF*7$Fdv$!3$ecP/UDtU$F*7$Fiv$!3dhAqr!3L_$F *7$F^w$!36AEoElNSOF*7$Fcw$!3'3G#*[&)4Cw$F*7$Fhw$!3bW()4oRx-RF*7$F]x$!3 18i96')4`SF*7$Fbx$!3#)z@o.@*[A%F*7$Fgx$!3?'\\%*GKm[S%F*7$F\\y$!3$G`A$) y(G/YF*7$Fay$!3St4))es4=[F*7$Ffy$!36BR&H+&GH]F*7$F[z$!3Atm)*f\")[*G&F* 7$F`z$!3==*fuYI%RbF*7$Fez$!3GX&4L'QsCeF*7$Fjz$!3o,hfHESofF*7$F_[l$!3Qr mN+R'o6'F*7$Fd[l$!3)>CUd0t=G'F*7$Fi[lF\\]lF[\\lFd\\l-F$6&7$7$Fb\\l$!37 *)RYv7b0OF*7$Fb\\l$\"37*)RYv7b0OF*-F\\\\l6&F^\\lFb\\l$\"*++++\"!\")Fb \\l-%'SYMBOLG6$%'CIRCLEGF`\\l-%&STYLEG6#%&POINTG-F$6&FjflFagl-Fggl6$%( DIAMONDGF`\\lFjgl-F$6&FjflFagl-Fggl6$%&CROSSGF`\\lFjgl-F$6&Fjfl-F\\\\l 6&F^\\lFc\\lFc\\lFc\\l-Fggl6$Figl\"#7Fjgl-F$6$7%F[gl7$$\"3/+++u[w'R$F* $\"3(********oUF\"fF*F^gl-%&COLORG6&F^\\lFb\\l$\"\")Fa\\lFb\\l-F$6&7#F bilFfglFjhlFjgl-F$6&F^jlF`hlFjhlFjgl-F$6&F^jlFehlFjhlFjgl-%%TEXTG6&7$$ !#&)!\"#$!#VFa\\lQ)F~(0,-c)6\"-Fhil6&F^\\l$\"\"\"FijlF`[mF`[m-%%FONTG6 $%*HELVETICAGF`\\l-Fdjl6&7$$\"#[Fa\\l$\"#eFa\\lQ'P(x,y)F][mF^[mFb[m-Fd jl6&7$Fgjl$\"#VFa\\lQ(F~(0,c)F][mF^[mFb[m-Fdjl6&7$$!#8Fa\\l$!#XFa\\lQ \"1F][mF^[m-Fc[m6$Fe[mF[jl-Fdjl6&7$$!$D\"Fijl$\"$0%FijlQ\"2F][mF^[mF\\ ]m-Fdjl6&7$$\"#\\Fa\\l$!\"$Fa\\lQ\"xF][mF^[m-Fc[m6$Fe[m\"\"*-Fdjl6&7$F [^m$\"#lFa\\lQ\"yF][mF^[mF^^m-%(SCALINGG6#%,CONSTRAINEDG-%*AXESTICKSG6 $Fc\\lFc\\lF^^m-%+AXESLABELSG6%Q!F][mFa_m-Fc[m6#%(DEFAULTG-%%VIEWG6$Fd _mFd_m" 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" }}{TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 "This hyperbola has the equation: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/(a^2)-x^2/(b^2) = 1;" "6#/,&*&%\"yG \"\"#*$%\"aGF'!\"\"\"\"\"*&%\"xGF'*$%\"bGF'F*F*F+" }{TEXT -1 17 " --- ---- (iii), " }}{PARA 0 "" 0 "" {TEXT -1 6 "where " }{XPPEDIT 18 0 "b \+ = sqrt(c^2-a^2);" "6#/%\"bG-%%sqrtG6#,&*$%\"cG\"\"#\"\"\"*$%\"aGF+!\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 19 "The hyperbola has \+ " }{TEXT 270 1 "y" }{TEXT -1 13 " intercepts " }{XPPEDIT 18 0 "y =`` " "6#/%\"yG%!G" }{TEXT 271 1 "+" }{TEXT -1 1 " " }{TEXT 280 1 "a" } {TEXT -1 9 ", but no " }{TEXT 281 1 "x" }{TEXT -1 14 " intercepts. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 18 "As befo re we have " }{XPPEDIT 18 0 "b^2 = c^2-a^2;" "6#/*$%\"bG\"\"#,&*$%\"cG F&\"\"\"*$%\"aGF&!\"\"" }{TEXT -1 14 ", which gives " }{XPPEDIT 18 0 " c^2 = a^2+b^2;" "6#/*$%\"cG\"\"#,&*$%\"aGF&\"\"\"*$%\"bGF&F*" }{TEXT -1 10 ", so that " }{XPPEDIT 18 0 "c = sqrt(a^2+b^2);" "6#/%\"cG-%%sqr tG6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF+F," }{TEXT -1 2 ". " }}{PARA 0 "" 0 " " {TEXT -1 28 "This means that the foci are" }{XPPEDIT 18 0 "``(0,sqrt (a^2+b^2));" "6#-%!G6$\"\"!-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF-F." }{TEXT -1 4 " and" }{XPPEDIT 18 0 "``(0,-sqrt(a^2+b^2));" "6#-%!G6$\" \"!,$-%%sqrtG6#,&*$%\"aG\"\"#\"\"\"*$%\"bGF.F/!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 43 "It is u seful to solve the equation (i) for " }{TEXT 292 1 "y" }{TEXT -1 13 " \+ in terms of " }{TEXT 293 1 "x" }{TEXT -1 2 ". " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/(a^2) = x^2/(b^2)+1;" "6#/*&%\"yG\" \"#*$%\"aGF&!\"\",&*&%\"xGF&*$%\"bGF&F)\"\"\"F/F/" }{TEXT -1 2 ", " }} {PARA 0 "" 0 "" {TEXT -1 4 "and " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y^2 = a^2*``(x^2/(b^2)+1);" "6#/*$%\"yG\"\"#*&%\"aGF&-% !G6#,&*&%\"xGF&*$%\"bGF&!\"\"\"\"\"F2F2F2" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 6 "Hence " }}{PARA 256 "" 0 "" {TEXT -1 1 " " } {XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT 294 1 "+" }{TEXT -1 1 " " } {XPPEDIT 18 0 "a*sqrt(x^2/(b^2)+1);" "6#*&%\"aG\"\"\"-%%sqrtG6#,&*&%\" xG\"\"#*$%\"bGF,!\"\"F%F%F%F%" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 5 "When " }{TEXT 295 1 "x" }{TEXT -1 46 " is large and positi ve, or large and negative " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{TEXT 297 1 "y" }{TEXT -1 1 " " }{TEXT 296 1 "~" }{TEXT -1 1 " " }{TEXT 298 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a/b;" "6#*&%\"aG\"\"\"%\"bG!\"\" " }{TEXT -1 1 " " }{TEXT 299 1 "x" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 66 "This means that the hyperbola becomes close to the straig ht lines " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y = a/b; " "6#/%\"yG*&%\"aG\"\"\"%\"bG!\"\"" }{TEXT -1 1 " " }{TEXT 300 1 "x" } {TEXT -1 9 " and " }{XPPEDIT 18 0 "y = -a/b;" "6#/%\"yG,$*&%\"aG\" \"\"%\"bG!\"\"F*" }{TEXT -1 1 " " }{TEXT 301 1 "x" }{TEXT -1 2 ". " }} {PARA 0 "" 0 "" {TEXT -1 16 "These lines are " }{TEXT 259 10 "asymptot es" }{TEXT -1 33 " for the graph of the hyperbola. 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+]<#Rm\\&F*7$F^hm$!3F++]A_ERdF*7$FchmFeilFjhlFghm-F$6%7'7$$Fg\\lFc\\l$ FjhmFc\\l7$$!\"#Fc\\lFbbn7$Fdbn$!\"$Fc\\l7$FabnFgbnF`bn-F\\\\l6&F^\\l$ \")!\\DP\"FeglF\\cn$\")viobFegl-FhhmFf\\l-%%TEXTG6&7$$\"\"&Fa\\l$!#OFa \\lQ\"F6\"-%&COLORG6&F^\\l$\"\"\"FebnF^dnF^dn-%%FONTG6$%*HELVETICAGF` \\l-Fbcn6&7$Fecn$\"#PFa\\lFicnF[dnF`dn-Fbcn6&7$$FebnFa\\l$\"\"(Fc\\lQ \"yFjcnF[dn-Fadn6$Fcdn\"\"*-Fbcn6&7$$\"#XFa\\lF\\enQ\"xFjcnF[dnF`en-Fb cn6&7$$FhbnFa\\l$!+v7b0O!\"*Q#-cFjcnF[dnF`en-Fbcn6&7$F\\fn$\"+v7b0OF_f nQ\"cFjcnF[dnF`en-Fbcn6&7$F\\en$!$w#FebnQ#-aFjcnF[dnF`en-Fbcn6&7$F\\en $\"$&GFebnQ\"aFjcnF[dnF`en-Fbcn6&7$FdbnF\\fnQ#-bFjcnF[dnF`en-Fbcn6&7$F abnF\\fnQ\"bFjcnF[dnF`en-Fbcn6&7$$FfhmFa\\l$!$&QFebnQ\"1FjcnF[dn-Fadn6 $Fcdn\"\")-Fbcn6&7$F^hn$\"#NFa\\lQ\"2FjcnF[dnFbhn-%*AXESTICKSG6$Fc\\lF c\\l-%+AXESLABELSG6%Q!FjcnFain-Fadn6#%(DEFAULTGF`en-%(SCALINGG6#%,CONS TRAINEDG-%%VIEWG6$;FcilFchmFdin" 1 2 0 1 10 0 2 9 1 4 1 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" "Curve 4" "Curve 5" "Curve 6" "Curve 7" "Curve 8" "Curve 9" "Curve 10" "Curve 11" "Cur ve 12" "Curve 13" "Curve 14" "Curve 15" "Curve 16" "Curve 17" "Curve 1 8" "Curve 19" "Curve 20" "Curve 21" }}{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 130 "The equations of the asymptotes can be obtained quickly \+ from equation (iii) by replacing the 1 on the right side by 0 to obtai n: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/(a^2)-x^2 /(b^2) = 0;" "6#/,&*&%\"yG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"xGF'*$%\"bGF'F *F*\"\"!" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 32 "This equatio n is equivalent to: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y^2/(a^2) = x^2/(b^2);" "6#/*&%\"yG\"\"#*$%\"aGF&!\"\"*&%\"xGF&*$% \"bGF&F)" }{TEXT -1 2 ", " }}{PARA 0 "" 0 "" {TEXT -1 14 "which gives: " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "y =``" "6#/%\"y G%!G" }{TEXT 313 1 "+" }{TEXT -1 1 " " }{XPPEDIT 18 0 "a/b;" "6#*&%\"a G\"\"\"%\"bG!\"\"" }{TEXT -1 1 " " }{TEXT 314 1 "x" }{TEXT -1 2 ". " } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 4 " " } }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examples " }}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Exam ple 1 " }}{PARA 0 "" 0 "" {TEXT 262 8 "Question" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 64 "(a) Find the focus and directrix of the p arabola with equation " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"# " }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 38 "(b) Sketch the graph of the parabola " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" } {TEXT -1 46 ", indicating clearly the focus and directrix. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 8 "Solution" } {TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 27 "(a) Comparing the equat ion " }{XPPEDIT 18 0 "y=x^2" "6#/%\"yG*$%\"xG\"\"#" }{TEXT -1 30 " wit h the \"standard\" equation " }{XPPEDIT 18 0 "x^2=4*a*y" "6#/*$%\"xG\" \"#*(\"\"%\"\"\"%\"aGF)%\"yGF)" }{TEXT -1 10 ", we have " }{XPPEDIT 18 0 "4*a=1" "6#/*&\"\"%\"\"\"%\"aGF&F&" }{TEXT -1 14 ", which gives \+ " }{XPPEDIT 18 0 "a=1/4" "6#/%\"aG*&\"\"\"F&\"\"%!\"\"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 43 " It follows that the focus is th e point" }{XPPEDIT 18 0 "``(0,1/4)" "6#-%!G6$\"\"!*&\"\"\"F(\"\"%!\"\" " }{TEXT -1 34 ", while the directrix is the line " }{XPPEDIT 18 0 "y= -1/4" "6#/%\"yG,$*&\"\"\"F'\"\"%!\"\"F)" }{TEXT -1 2 ". " }}{PARA 0 " " 0 "" {TEXT -1 4 "(b) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 " p1 := plot(x^2,x=-1.5..1.5,thickness=2):\np2 := plot(-1/4,x=-1.5..1.5, color=blue,thickness=2):\np3 := plot([[[0,1/4]]$3],style=point,color=b lack,symbol=[circle,diamond,cross]):\nplots[display]([p1,p2,p3],scalin g=constrained,tickmarks=[3,3]);" }}{PARA 13 "" 1 "" {GLPLOT2D 283 236 236 {PLOTDATA 2 "6+-%'CURVESG6%7S7$$!3++++++++:!#<$\"3+++++++]AF*7$$!3 &*****\\P&3YV\"F*$\"3i))y'el,\"e?F*7$$!3!***\\iv$FX7$$!3')*****\\FPm( \\FX$\"3UU>pc=pwCFX7$$!3()*******4'*QS%FX$\"3%4_ff3I%R>FX7$$!3?++Dc>mP PFX$\"3'y&)>**o6qR\"FX7$$!3'3+++&=$z9$FX$\"3bFWCK\\Z4**!#>7$$!3N***\\i X/4]#FX$\"3C06H*4BXD'Fgq7$$!3C***\\(o8y%)=FX$\"3)47()z!3S_NFgq7$$!33** **\\i:#>C\"FX$\"3#!#B7$$\"3i(***\\P>:mkFgq$\"3?(\\$)y3 76=%F\\s7$$\"3d***\\iv&QA7FX$\"3S$)G3PpA%\\\"Fgq7$$\"3j++]PPBW=FX$\"3J @K`y!)>,MFgq7$$\"3%*)*****\\Nm'[#FX$\"3'pf)*3h&\\$='Fgq7$$\"36****\\(y b^6$FX$\"3SRZR!e&>/(*Fgq7$$\"3')***\\PMaKs$FX$\"3;C`#3HiiQ\"FX7$$\"3a* ***\\7TW)R%FX$\"3z!fy51JY$>FX7$$\"3*y*****\\@80]FX$\"3-gj*QyM^]#FX7$$ \"3_+++D6!Hl&FX$\"3)pi-H6Hb>$FX7$$\"3j)**\\P4w)RiFX$\"3Mu_`m`g$*QFX7$$ \"3s,++vZf\")oFX$\"3OK2tkYjNZFX7$$\"3'z**\\P/-a[(FX$\"3IkwlvV7.cFX7$$ \"3R++v=Yb;\")FX$\"3)o>:z)e%ye'FX7$$\"3s)****\\i@Ot)FX$\"3Jin')oYhFwFX 7$$\"3g)**\\PfL'z$*FX$\"3+^.INEv(z)FX7$$\"3>+++!*>=+5F*$\"3Qg.78SO+5F* 7$$\"3-++DE&4Q1\"F*$\"3T\\-932pJ6F*7$$\"3=+]P%>5p7\"F*$\"3s+j='eE*p7F* 7$$\"39+++bJ*[=\"F*$\"3Faew)yrRS\"F*7$$\"33++Dr\"[8D\"F*$\"3[>2pXA(ec \"F*7$$\"3++++Ijy58F*$\"3%*o[\"H!3;= " 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 -1 115 "Find the equation of the parabola which has its focus at the point ``(-1,1) and has the line y =x as its directrix. " }}{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 45 "Code for animations of hyperbola construction" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 918 "aa := 3: bb := 2: cc := sqrt(13):\np1 := plot([aa*cosh(t),bb*sinh (t),t=-1.4..1.4],thickness=2):\np2 := plot([-aa*cosh(t),bb*sinh(t),t=- 1.4..1.4],thickness=2):\np3 := plot([[[-cc,0],[cc,0]]$4],style=point,s ymbol=[circle,diamond,cross,circle],\n color=[green$3,black],symbols ize=[10$3,12]):\ntt := 1:\np4 := plot([[-cc,0],[aa*cosh(tt),bb*sinh(tt )],[cc,0]],color=COLOR(RGB,0,.8,0)):\np5 := plot([[[aa*cosh(tt),bb*sin h(tt)]]$3],style=point,symbol=[circle,diamond,cross],\n color=black ,symbolsize=[10$3]):\nt1 := plots[textplot]([[6.6,-.25,`x`],[-.25,4.59 ,`y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := p lots[textplot]([[-4.1,.5,`F`],[4.15,.5,`F`]],font=[TIMES,BOLD,11],\n \+ color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[-3.9,.3,`1`], [4.35,.3,`2`]],font=[HELVETICA,8],\n color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p||(1..5),t1,t2,t3],font=[HELVETICA,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 1350 "aa := 3: bb := 2: cc := sqrt(13):\nnumframes := 40:\np1 := plot( [[[-cc,0],[cc,0]]$4],style=point,symbol=[circle,diamond,cross,circle], \n color=[green$3,black],symbolsize=[10$3,12]):\nt1 := plots[textplo t]([[6.6,-.25,`x`],[-.25,4.59,`y`]],font=[HELVETICA,9],\n color=COLO R(RGB,.01,.01,.01)):\nfrms := NULL:\nnn := numframes-1:\ndd := evalf(2 .8/nn):\nfor i from 0 to nn do\n tt := -1.4+dd*i;\n p2 := plot([[- cc,0],[aa*cosh(tt),bb*sinh(tt)],[cc,0]],color=COLOR(RGB,0,.8,0)):\n \+ p3 := plot([[[aa*cosh(tt),bb*sinh(tt)]]$3],style=point,symbol=[circle, diamond,cross],\n color=black,symbolsize=[10$3]):\n if i=0 t hen\n p4 := plot([aa*cosh(t),bb*sinh(t),t=-1.4..1.4],color=COLOR( RGB,.5,.5,.5),\n linestyle=1,thickness =1):\n frms := frms,plots[display]([p||(1..4),t1]); \n elif i =nn then\n p4 := plot([aa*cosh(t),bb*sinh(t),t=-1.4..1.4],color=r ed,thickness=3):\n frms := frms,plots[display]([p||(1..4),t1]);\n else \n p4 := plot([aa*cosh(t),bb*sinh(t),t=-1.4..tt],col or=red,thickness=3):\n p5 := plot([aa*cosh(t),bb*sinh(t),t=tt..1. 4],color=COLOR(RGB,.5,.5,.5),\n linest yle=1,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 scaling=constrained,tickmarks=[6,4]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1357 "aa \+ := 3: bb := 2: cc := sqrt(13):\nnumframes := 40:\np1 := plot([[[-cc,0] ,[cc,0]]$4],style=point,symbol=[circle,diamond,cross,circle],\n colo r=[green$3,black],symbolsize=[10$3,12]):\nt1 := plots[textplot]([[2.59 ,-.25,`x`],[-.25,4.59,`y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.0 1,.01,.01)):\nfrms := NULL:\nnn := numframes-1:\ndd := evalf(2.8/nn): \nfor i from 0 to nn do\n tt := -1.4+dd*i;\n p2 := plot([[-cc,0],[ -aa*cosh(tt),bb*sinh(tt)],[cc,0]],color=COLOR(RGB,0,.8,0)):\n p3 := \+ plot([[[-aa*cosh(tt),bb*sinh(tt)]]$3],style=point,symbol=[circle,diamo nd,cross],\n color=black,symbolsize=[10$3]):\n if i=0 then\n p4 := plot([-aa*cosh(t),bb*sinh(t),t=-1.4..1.4],color=COLOR(RGB, .5,.5,.5),\n linestyle=1,thickness=1): \n frms := frms,plots[display]([p||(1..4),t1]); \n elif i=nn \+ then\n p4 := plot([-aa*cosh(t),bb*sinh(t),t=-1.4..1.4],color=red, thickness=3):\n frms := frms,plots[display]([p||(1..4),t1]);\n \+ else \n p4 := plot([-aa*cosh(t),bb*sinh(t),t=-1.4..tt],color =red,thickness=3):\n p5 := plot([-aa*cosh(t),bb*sinh(t),t=tt..1.4 ],color=COLOR(RGB,.5,.5,.5),\n linesty le=1,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 scaling=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 28 "General hyperbola pictures. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 946 "aa := 3: bb := 2: cc := sqrt(13):\np1 := plot([aa*cosh(t),bb*sinh (t),t=-1.4..1.4],thickness=2):\np2 := plot([-aa*cosh(t),bb*sinh(t),t=- 1.4..1.4],thickness=2):\np3 := plot([[[-cc,0],[cc,0]]$4],style=point,s ymbol=[circle,diamond,cross,circle],\n color=[green$3,black],symbols ize=[10$3,12]):\ntt := 1.3:\np4 := plot([[-cc,0],[aa*cosh(tt),bb*sinh( tt)],[cc,0]],color=COLOR(RGB,0,.8,0)):\np5 := plot([[[aa*cosh(tt),bb*s inh(tt)]]$3],style=point,symbol=[circle,diamond,cross],\n color=bla ck,symbolsize=[10$3]):\nt1 := plots[textplot]([[-4.1,-.4,`F (-c,0)`],[ 5.2,4.1,`P(x,y)`],[4.1,-.4,`F (c,0)`]],font=[HELVETICA,10],color=COLOR (RGB,.01,.01,.01)):\nt2 := plots[textplot]([[-4.53,-.6,`1`],[3.73,-.6, `2`]],font=[HELVETICA,8],\n color=COLOR(RGB,.01,.01,.01)):\nt3 := pl ots[textplot]([[7,-.2,`x`],[-.2,4.5,`y`]],font=[HELVETICA,9],\n colo r=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5),t1,t2,t3],font=[ HELVETICA,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 940 "aa : = 3: bb := 2: cc := sqrt(13):\np1 := plot([aa*cosh(t),bb*sinh(t),t=-1. 5..1.5],thickness=2):\np2 := plot([-aa*cosh(t),bb*sinh(t),t=-1.5..1.5] ,thickness=2):\np3 := plot([[[-cc,0],[cc,0]]$4],style=point,symbol=[ci rcle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3 ,12]):\np4 := plot([bb/aa*x,-bb/aa*x],x=-7..7,color=black,linestyle=3) :\np5 := plot([[aa,bb],[-aa,bb],[-aa,-bb],[aa,-bb],[aa,bb]],color=navy ,linestyle=2):\nt1 := plots[textplot]([[-3.7,.5,`F`],[3.7,.5,`F`]],fon t=[HELVETICA,10],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[te xtplot]([[7,-.2,`x`],[-.2,4.5,`y`],[-cc,-.3,`-c`],[cc,-.3,`c`],\n \+ [-2.82,-.3,`-a`],[2.85,-.3,`a`],[-.3,-2,`-b`],[-.3,2,`b`]],\n font= [HELVETICA,9],color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[ -3.5,.4,`1`],[3.9,.4,`2`]],font=[HELVETICA,8],\n color=COLOR(RGB,.01 ,.01,.01)):\nplots[display]([p||(1..5),t1,t2,t3],font=[HELVETICA,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 952 "aa := 3: bb := 2: cc := sqrt(13):\np1 := plot([bb*sinh(t),aa*cosh(t),t=-1.4..1.4],thick ness=2):\np2 := plot([bb*sinh(t),-aa*cosh(t),t=-1.4..1.4],thickness=2) :\np3 := plot([[[0,-cc],[0,cc]]$4],style=point,symbol=[circle,diamond, cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\ntt := 1.3:\np4 := plot([[0,-cc],[bb*sinh(tt),aa*cosh(tt)],[0,cc]],color=COL OR(RGB,0,.8,0)):\np5 := plot([[[bb*sinh(tt),aa*cosh(tt)]]$3],style=poi nt,symbol=[circle,diamond,cross],\n color=black,symbolsize=[10$3]): \nt1 := plots[textplot]([[-.85,-4.3,`F (0,-c)`],[4.8,5.8,`P(x,y)`],[-. 85,4.3,`F (0,c)`]],font=[HELVETICA,10],color=COLOR(RGB,.01,.01,.01)): \nt2 := plots[textplot]([[-1.3,-4.5,`1`],[-1.25,4.05,`2`]],font=[HELVE TICA,8],\n color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[4 .9,-.3,`x`],[-.3,6.5,`y`]],font=[HELVETICA,9],\n color=COLOR(RGB,.01 ,.01,.01)):\nplots[display]([p||(1..5),t1,t2,t3],font=[HELVETICA,9],\n scaling=constrained,tickmarks=[0,0]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 941 "aa := 3: bb := 2: cc := sqrt(13):\np1 := plot([bb*sinh(t),aa*cosh(t),t=-1.5..1.5],thick ness=2):\np2 := plot([bb*sinh(t),-aa*cosh(t),t=-1.5..1.5],thickness=2) :\np3 := plot([[[0,-cc],[0,cc]]$4],style=point,symbol=[circle,diamond, cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\np4 := plot([aa/bb*x,-aa/bb*x],x=-4..4,color=black,linestyle=3):\np5 := plot ([[bb,aa],[-bb,aa],[-bb,-aa],[bb,-aa],[bb,aa]],color=navy,linestyle=2) :\nt1 := plots[textplot]([[.5,-3.6,`F`],[.5,3.7,`F`]],font=[HELVETICA, 10],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[-.2 ,7,`y`],[4.5,-.2,`x`],[-.3,-cc,`-c`],[-.3,cc,`c`],\n [-.2,-2.76,`- a`],[-.2,2.85,`a`],[-2,-.3,`-b`],[2,-.3,`b`]],\n font=[HELVETICA,9] ,color=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[.6,-3.85,`1`] ,[.6,3.5,`2`]],font=[HELVETICA,8],\n color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p||(1..5),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 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 }