{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 "" -1 262 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 1 0 0 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 1 0 0 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 "" 258 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 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 } {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 "Time s" 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 "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 61 "Rectangular hyperbolas with the c oordinate axes as asymptotes" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter \+ Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 18 "Version: \+ 24.3.2007" }}{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 50 "Rotating a rectangular hype rbola about the origin " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 78 "A hyperbola with asymptotes which are perpendicular \+ to each other is called a " }{TEXT 259 21 "rectangular hyperbola" } {TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 29 "The hyperbola with equa tion " }{XPPEDIT 18 0 "x^2-y^2 = 2;" "6#/,&*$%\"xG\"\"#\"\"\"*$%\"yGF '!\"\"F'" }{TEXT -1 61 " is a rectangular hyperbola. It has the perpe ndicular lines " }{XPPEDIT 18 0 "y=``" "6#/%\"yG%!G" }{TEXT 262 1 "+" }{TEXT -1 1 " " }{TEXT 263 1 "x" }{TEXT -1 47 " as its asymptotes, and the foci are the points" }{XPPEDIT 18 0 "``(-2,0)" "6#-%!G6$,$\"\"#! \"\"\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "``(2,0)" "6#-%!G6$\"\"# \"\"!" }{TEXT -1 2 ". 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" }}{PARA 0 "" 0 "" {TEXT -1 61 " Check that the differen ce of the distances of the point" }{XPPEDIT 18 0 "``(4,1/2)" "6#-%!G6$ \"\"%*&\"\"\"F(\"\"#!\"\"" }{TEXT -1 50 " from the two foci is 4. Do t he same for the point" }{XPPEDIT 18 0 "``(2, 1);" "6#-%!G6#6$\"\"#\"\" \"" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 107 "(b) Find the equa tion of rectangular hyperbola that has its centre at the origin and fo ci which are on the " }{TEXT 266 1 "x" }{TEXT -1 172 " axis and are th e same distance apart as the foci of the hyperbola in (a) and such tha t the difference of the distances of any point on the hyperbola rom th e two foci is 4." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 " (a) " }}{PARA 256 "" 0 "" {TEXT -1 1 " " }{GLPLOT2D 381 385 385 {PLOTDATA 2 "63-%'CURVESG6&7gn7$$!3k*************f'!#<$!3RIIIIIIII!#=7 $$!3!>s(y#yQhX'F*$!3u4tORo#y4$F-7$$!3:oI\"p!f'4L'F*$!3i!>'F*$!3LE'>0^:4B$F-7$$!3k8IN6G\\[gF*$!3nD;Gt)3mI$F-7$$!3*H2g i>ju!fF*$!3kD>fkza&Q$F-7$$!37y:-!p5nx&F*$!3;7p;&)yN#HVSKTc F*$!3G&H^e+n_a$F-7$$!3Y&Qs&zpI,bF*$!33I@$Hs*\\NOF-7$$!3)>_^B&*Q]QF*$!3S/2.&F-7$$!3!ep&*o!>l9PF*$!3!f\\0gc%3%Q&F-7$$!3E aN!)\\FAtNF*$!3%y_,49)=(f&F-7$$!3L1$)=MmMVMF*$!3]$=#**HGI3eF-7$$!3m%[V 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}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 18 "Code for pictures " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 0 "" 0 " " {TEXT -1 21 "Rectangular hyperbola" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 868 "aa := evalf(sqrt(2)):\np1 := plot([aa*cosh(t),aa*sinh(t),t=-1.82..1.82],thickness=2):\np2 := pl ot([-aa*cosh(t),aa*sinh(t),t=-1.82..1.82],thickness=2):\np3 := plot([[ [-2,0],[2,0]]$4],style=point,symbol=[circle,diamond,cross,circle],\n \+ color=[green$3,black],symbolsize=[10$3,12]):\np4 := plot([x,-x],x=-4. 5..4.5,color=brown,linestyle=1):\np5 := plot([[aa,aa],[-aa,aa],[-aa,-a a],[aa,-aa],[aa,aa]],color=navy,linestyle=2):\nt1 := plots[textplot]([ [-2.25,.35,`F`],[2.1,.35,`F`]],font=[HELVETICA,10],\n color=COLOR( RGB,.01,.01,.01)):\nt2 := plots[textplot]([[4.7,-.19,`x`],[-.19,4.7,`y `]],\n font=[HELVETICA,9],color=COLOR(RGB,.01,.01,.01)):\nt3 := plo ts[textplot]([[-2.13,.24,`1`],[2.22,.24,`2`]],font=[HELVETICA,8],\n \+ color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..5),t1,t2,t3],fo nt=[HELVETICA,9],\n scaling=constrained,tickmarks=[5,5],view=[-4.5. .4.7,-4.5..4.7]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 782 "aa := evalf(sqrt(2)):\np1 := plot(1/x,x=-6. .6,thickness=2,discont=true):\np2 := plot([[[-aa,-aa],[aa,aa]]$4],styl e=point,symbol=[circle,diamond,cross,circle],\n color=[green$3,black ],symbolsize=[10$3,12]):\np3 := plot([[2,0],[0,2],[-2,0],[0,-2],[2,0]] ,color=navy,linestyle=2):\np4 := plot([x,-x],x=-6..6,color=black,lines tyle=3):\nt1 := plots[textplot]([[-1.74,-1.18,`F`],[1.26,1.87,`F`]],fo nt=[HELVETICA,10],\n color=COLOR(RGB,.01,.01,.01)):\nt2 := plots[t extplot]([[6.6,-.19,`x`],[-.19,6.6,`y`]],\n font=[HELVETICA,9],colo r=COLOR(RGB,.01,.01,.01)):\nt3 := plots[textplot]([[-1.6,-1.34,`1`],[1 .39,1.73,`2`]],font=[HELVETICA,8],\n color=COLOR(RGB,.01,.01,.01)): \nplots[display]([p||(1..4),t1,t2,t3],font=[HELVETICA,9],\n scaling =constrained,tickmarks=[5,5],view=[-6..6.6,-6..6.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 0 "" 0 "" {TEXT -1 35 "Rotation of rectangular hyperbola " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1024 "R := t -> ((x,y) -> (cos(t)*x-sin(t)*y,sin(t)*x+cos (t)*y)):\nLL := (x,y)->[x,y]:\naa := evalf(sqrt(2)):\npp := plot([aa*c osh(t),aa*sinh(t),t=-1.82..1.82]):\npts1 := op(1,op(1,pp)):\npp := plo t([-aa*cosh(t),aa*sinh(t),t=-1.82..1.82]):\npts2 := op(1,op(1,pp)):\np p := plot(x,x=-4.5..4.5,adaptive=false,numpoints=10):\npts3 := op(1,op (1,pp)):\npp := plot(-x,x=-4.5..4.5,adaptive=false,numpoints=10):\npts 4 := op(1,op(1,pp)):\nmap(LL@R(tt)@op,pts):\nt1 := plots[textplot]([[6 .6,-.22,`x`],[-.22,6.6,`y`]],\n font=[HELVETICA,9],color=COLOR(RGB, .01,.01,.01)):\nnumframes := 40:\nfrms := NULL:\nnn := numframes-1:\nd d := evalf(Pi/(4*nn)):\nfor i from 0 to nn do\n tt := dd*i;\n RL : = (LL@R(tt)@op);\n p1 := plot(map(RL,pts1),color=red,thickness=2);\n p2 := plot(map(RL,pts2),color=red,thickness=2);\n p3 := plot(map( RL,pts3),color=brown,thickness=1);\n p4 := plot(map(RL,pts4),color=b rown,thickness=1);\n frms := frms,plots[display]([p||(1..4),t1]);\ne nd do:\nplots[display]([frms],insequence=true,scaling=constrained,view =[-6..6.6,-6..6.6]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 0 "" 0 "" {TEXT -1 24 "Pictures for the example" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 708 "aa := evalf(sqrt(2)):\np1 : = plot(2/x,x=-6.6..6.6,thickness=2,discont=true):\np2 := plot([[[-2,-2 ],[2,2],[4,.5]]$4],style=point,symbol=[circle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3,12]):\np3 := plot(x,x=-6..6, color=black,linestyle=3):\np4 := plot([[-2,-2],[4,.5],[2,2]],color=blu e,thickness=2):\nt1 := plots[textplot]([[-2.4,-1.4,`(-2,-2)`],[1.7,2.6 ,`(2,2)`],\n [4.4,1,`(4,1/2)`]],font=[HELVETICA,10],\n col or=COLOR(RGB,.01,.01,.01)):\nt2 := plots[textplot]([[6.6,-.25,`x`],[-. 25,6.6,`y`]],\n font=[HELVETICA,9],color=COLOR(RGB,.01,.01,.01)):\n plots[display]([p||(1..4),t1,t2],font=[HELVETICA,9],\n scaling=cons trained,tickmarks=[5,5],labels=[``,``],view=[-6..6.6,-6..6.6]);" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 560 "aa := evalf(2*sqrt(2)):\np1 := plot([2*cosh(t),2*sinh(t),t=-1.82. .1.82],thickness=2):\np2 := plot([-2*cosh(t),2*sinh(t),t=-1.82..1.82], thickness=2):\np3 := plot([[[-aa,0],[aa,0]]$4],style=point,symbol=[cir cle,diamond,cross,circle],\n color=[green$3,black],symbolsize=[10$3, 12]):\np4 := plot([x,-x],x=-4.5..4.5,color=black,linestyle=2):\nt1 := \+ plots[textplot]([[4.7,-.19,`x`],[-.19,4.7,`y`]],\n font=[HELVETICA, 9],color=COLOR(RGB,.01,.01,.01)):\nplots[display]([p||(1..4),t1],font= [HELVETICA,9],\n scaling=constrained,tickmarks=[5,5],view=[-4.5..4. 7,-4.5..4.7]);" }}}{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 }