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::xI::::::::::::::bMnCJA;W\\Hlysy:::::::::::::::::::::::vPt>W:RlvYxI:::::::::::::::::::::::i?nCjysy:ZqryvY::::::::::::::::: ::tBbMnGxIyA:vPJa:yay=::::::::::::::::::::::::::t@W\\H;:::::JqryvY:::: :::::::::::::::::::::::::::::::::::::::::::::::::::tBbMnGjy;:::::::::: :::::::W>nCt@ry::vPJa:yay=:::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::<:ry::::::::::::::ZZ@NbM:Zymy;:::::::::::::::::::::::i? nCjysy:ZqryvY:::::::::::::::::::XGi_ql^;yA:vPt>W:::::b]vYxI::::::::::::::::::::::::::::::::::::::::: ::::::::::::::roZql^;yA:::::::::::::::::vTjUSlZy=:jUcMnCjysy:::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::Z::xI:::::::::::::::W \\H;JqryvY:::::::::::::::::::::::jU;W:1:" }{TEXT -1 0 "" }}{PARA 18 "" 0 "" {TEXT 357 27 "Caract \351risation de coniques" }}{PARA 18 "" 0 "" {TEXT 358 39 "\251 Appren dre Maple. Alain Le Stang. 2006" }{TEXT 356 1 "\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "restart:with(linalg):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 80 "Warning, the protected names norm and trace have \+ been redefined and unprotected\n" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warni ng, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 331 16 "M\351thode utilis\351e" }{TEXT 332 0 "" }}}{EXCHG {PARA 259 "" 0 "" {TEXT -1 113 "Soit dans un rep\350re orthonormal (o, i , j) du plan affine euclidien, une conique C d\351finie par son \+ \351quation :\nf (" }{TEXT 339 1 "x" }{TEXT -1 2 ", " }{TEXT 340 1 "y " }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "A*x^2+2*B*x*y+C*y^2+D*x+E*y+F = 0 ;" "6#/,.*&%\"AG\"\"\"*$%\"xG\"\"#F'F'**F*F'%\"BGF'F)F'%\"yGF'F'*&%\"C GF'*$F-F*F'F'*&%\"DGF'F)F'F'*&%\"EGF'F-F'F'%\"FGF'\"\"!" }}{PARA 259 " " 0 "" {TEXT -1 102 "les coefficients A, B, C, D, E, F \351tant r\351e ls. On rep\350re alors la forme quadratique de IR\262 associ\351e: " }{TEXT 343 1 "q" }{TEXT -1 2 "( " }{TEXT 344 1 "x" }{TEXT -1 2 ", " } {TEXT 341 1 "y" }{TEXT -1 4 ") = " }{XPPEDIT 18 0 "A*x^2+2*B*x*y+C*y^2 ;" "6#,(*&%\"AG\"\"\"*$%\"xG\"\"#F&F&**F)F&%\"BGF&F(F&%\"yGF&F&*&%\"CG F&*$F,F)F&F&" }{TEXT -1 2 " ." }}{PARA 259 "" 0 "" {TEXT -1 14 "La mat rice de " }{TEXT 342 1 "q" }{TEXT -1 42 " dans la base canonique ( i, \+ j ) s'\351crit " }{XPPEDIT 18 0 "M = matrix([[A, B], [B, C]]);" "6#/% \"MG-%'matrixG6#7$7$%\"AG%\"BG7$F+%\"CG" }{TEXT -1 103 " . M est une m atrice sym\351trique qui est diagonalisable \net il existe donc une ba se orthonormale B' = ( " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\" &F$6#\"\"#" }{TEXT -1 40 " ) de IR\262 form\351e de vecteurs propres d e " }{TEXT 345 1 "M" }{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT -1 141 "On peut donc diagonaliser la matrice dans une base orthonormale direc te de vecteurs propres. On note P la matrice de passage de ( i, j )\n \340 ( " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\"&F$6#\"\"#" } {TEXT -1 6 " ) et " }{XPPEDIT 18 0 "lambda;" "6#%'lambdaG" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "mu;" "6#%#muG" }{TEXT -1 40 " les deux valeur s propres de la matrice " }{TEXT 346 1 "M" }{TEXT -1 1 "." }}{PARA 259 "" 0 "" {TEXT -1 11 "La matrice " }{TEXT 347 1 "M" }{TEXT -1 23 " \+ est donc semblable \340 " }{XPPEDIT 18 0 "D = matrix([[lambda, 0], [0 , mu]]);" "6#/%\"DG-%'matrixG6#7$7$%'lambdaG\"\"!7$F+%#muG" }{TEXT -1 5 " et " }{TEXT 348 1 "q" }{TEXT -1 2 " (" }{XPPEDIT 18 0 "X*e[1]+Ye[ 2];" "6#,&*&%\"XG\"\"\"&%\"eG6#F&F&F&&%#YeG6#\"\"#F&" }{TEXT -1 5 " ) \+ = " }{XPPEDIT 18 0 "lambda*X^2+mu*Y^2;" "6#,&*&%'lambdaG\"\"\"*$%\"XG \"\"#F&F&*&%#muGF&*$%\"YGF)F&F&" }{TEXT -1 37 " . B' \351tant orthonor male et une base " }{TEXT 349 1 "q" }{TEXT -1 13 "-orthogonale." }} {PARA 259 "" 0 "" {TEXT -1 56 "Dans cette nouvelle base, on n'a donc p lus de termes en " }{TEXT 350 3 "xy." }}{PARA 259 "" 0 "" {TEXT -1 53 "On se ram\350ne ensuite \340 des formes canoniques connues " }}{PARA 259 "" 0 "" {TEXT 351 1 "-" }{TEXT -1 92 " Soit on trouve des coniques propres: paraboles, hyperboles et des ellipses (ou des cercles)" }} {PARA 259 "" 0 "" {TEXT 352 1 "-" }{TEXT -1 89 " Soit on trouve des co niques d\351g\351n\351r\351s : couple de droites, droite, point, ensem ble vide." }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 256 "" 0 "" {TEXT -1 47 "Premier cas : si 0 n'est pas valeur propre de M" }} {EXCHG {PARA 259 "" 0 "" {TEXT -1 32 "Dans ce cas, det(M) = ac - b\262 = " }{XPPEDIT 18 0 "lambda*mu <> 0;" "6#0*&%'lambdaG\"\"\"%#muGF&\"\" !" }{TEXT -1 49 ". Alors la conique C admet un centre de sym\351trie \+ " }{XPPEDIT 18 0 "Omega(x[0],y[0]);" "6#-%&OmegaG6$&%\"xG6#\"\"!&%\"yG 6#F)" }{TEXT -1 17 " dans (O, i , j) " }}{PARA 259 "" 0 "" {TEXT -1 4 "o\371 (" }{XPPEDIT 18 0 "x[0],y[0];" "6$&%\"xG6#\"\"!&%\"yG6#F&" } {TEXT -1 23 ") v\351rifie le syst\350me : " }{XPPEDIT 18 0 "Diff(f,x)* (x[0], y[0]);" "6#*&-%%DiffG6$%\"fG%\"xG\"\"\"6$&F(6#\"\"!&%\"yG6#F-F) " }{TEXT -1 9 " = 0 et " }{XPPEDIT 18 0 "Diff(f,y)*(x[0], y[0]);" "6# *&-%%DiffG6$%\"fG%\"yG\"\"\"6$&%\"xG6#\"\"!&F(6#F.F)" }{TEXT -1 4 " = \+ 0" }}{PARA 259 "" 0 "" {TEXT -1 23 "En effet, les formules " } {XPPEDIT 18 0 "X = x-x[0];" "6#/%\"XG,&%\"xG\"\"\"&F&6#\"\"!!\"\"" } {TEXT -1 4 " et " }{XPPEDIT 18 0 "Y = y-y[0];" "6#/%\"YG,&%\"yG\"\"\"& F&6#\"\"!!\"\"" }{TEXT -1 55 " permettent d'obtenir l'\351quation de l a conique C dans (" }{XPPEDIT 18 0 "Omega,i,j;" "6%%&OmegaG%\"iG%\"jG " }{TEXT -1 23 ") sous la forme : \nC : " }{XPPEDIT 18 0 "A*x^2+2*B*x* y+C*y^2;" "6#,(*&%\"AG\"\"\"*$%\"xG\"\"#F&F&**F)F&%\"BGF&F(F&%\"yGF&F& *&%\"CGF&*$F,F)F&F&" }{TEXT -1 43 " = a . Les termes en x et y dispar aissent." }}{PARA 259 "" 0 "" {TEXT -1 13 "Ainsi C : q( " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 5 "M) = " }{TEXT 354 1 "a" }{TEXT -1 30 ", s'\351crit sous la forme C : " }{XPPEDIT 18 0 "lambda*X^2+m u*Y^2;" "6#,&*&%'lambdaG\"\"\"*$%\"XG\"\"#F&F&*&%#muGF&*$%\"YGF)F&F&" }{TEXT -1 3 " = " }{TEXT 353 1 "a" }{TEXT -1 17 " dans le rep\350re ( " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\"&F$6#\"\"#" }{TEXT -1 3 " )." }}} {SECT 0 {PARA 256 "" 0 "" {TEXT -1 10 "Sous cas 1" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 49 "Si a = 0 alors C est la r\351union de 2 droites \+ si " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{XPPEDIT 18 0 "mu" "6#%#mu G" }{TEXT -1 3 " <0" }}{PARA 259 "" 0 "" {TEXT -1 26 " ou \+ C = \{ " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 33 " \} \+ si " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" } {XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 3 " >0" }}}}{SECT 0 {PARA 258 " " 0 "" {TEXT 333 10 "Sous cas 2" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "Si " }{XPPEDIT 18 0 "a <> 0;" "6#0%\"aG\"\"!" }{TEXT -1 11 " alors \+ C : " }{XPPEDIT 18 0 "X^2/a*lambda+Y^2/a*mu = 1;" "6#/,&*(%\"XG\"\"#% \"aG!\"\"%'lambdaG\"\"\"F+*(%\"YGF'F(F)%#muGF+F+F+" }{TEXT -1 28 " . D istinguons alors 2 cas :" }}}{SECT 0 {PARA 258 "" 0 "" {TEXT 334 10 "S ous cas i" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "Si " }{XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{XPPEDIT 18 0 "mu" "6#%#muG" }{TEXT -1 16 " > \+ 0 , C = vide" }}{PARA 259 "" 0 "" {TEXT -1 24 " ou C \+ : " }{XPPEDIT 18 0 "X^2/(a^2)+Y^2/(b^2) = 1;" "6#/,&*&%\"XG\"\"#*$%\"a GF'!\"\"\"\"\"*&%\"YGF'*$%\"bGF'F*F+F+" }{TEXT -1 35 ", C est donc une ellipse de centre " }{XPPEDIT 18 0 "Omega" "6#%&OmegaG" }{TEXT -1 25 " (on a alors B\262 - AC < 0)" }}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 335 11 "Sous cas ii" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "Si " } {XPPEDIT 18 0 "lambda" "6#%'lambdaG" }{XPPEDIT 18 0 "mu" "6#%#muG" } {TEXT -1 11 " < 0 , C : " }{XPPEDIT 18 0 "X^2/(a^2)-Y^2/(b^2) = 1;" "6 #/,&*&%\"XG\"\"#*$%\"aGF'!\"\"\"\"\"*&%\"YGF'*$%\"bGF'F*F*F+" }{TEXT -1 37 ", C est donc une hyperbole de centre " }{XPPEDIT 18 0 "Omega" " 6#%&OmegaG" }{TEXT -1 25 " (on a alors B\262 - AC > 0)" }}}}}}{SECT 0 {PARA 258 "" 0 "" {TEXT 336 40 "Second cas : si 0 est valeur propre de M" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 31 "Dans ce cas det(M) = ac - \+ b\262 = " }{XPPEDIT 18 0 "lambda*mu = 0;" "6#/*&%'lambdaG\"\"\"%#muGF& \"\"!" }{TEXT -1 2 ". " }}{PARA 259 "" 0 "" {TEXT -1 15 "Si par exempl e " }{XPPEDIT 18 0 "lambda <> 0;" "6#0%'lambdaG\"\"!" }{TEXT -1 4 " et " }{XPPEDIT 18 0 "mu = 0;" "6#/%#muG\"\"!" }{TEXT -1 11 ", alors q ( " }{XPPEDIT 18 0 "X*e[1]+Ye[2];" "6#,&*&%\"XG\"\"\"&%\"eG6#F&F&F&&%#Ye G6#\"\"#F&" }{TEXT -1 6 " ) = " }{XPPEDIT 18 0 "lambda*X^2" "6#*&%'la mbdaG\"\"\"*$%\"XG\"\"#F%" }}{PARA 259 "" 0 "" {TEXT -1 19 "Dans le re p\350re (O, " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\"&F$6#\"\"# " }{TEXT -1 46 " ) l'\351quation de la conique C est de la forme " } {XPPEDIT 18 0 "lambda*X^2+D[1]*X+E[1]*Y+F[1] = 0;" "6#/,**&%'lambdaG\" \"\"*$%\"XG\"\"#F'F'*&&%\"DG6#F'F'F)F'F'*&&%\"EG6#F'F'%\"YGF'F'&%\"FG6 #F'F'\"\"!" }{TEXT -1 43 " que l'on peut mettre \ndans le rep\350re ( O, " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\"&F$6#\"\"#" }{TEXT -1 21 " ) sous la forme C : " }{XPPEDIT 18 0 "(X+a)^2+beta*(Y+gamma) = 0;" "6#/,&*$,&%\"XG\"\"\"%\"aGF(\"\"#F(*&%%betaGF(,&%\"YGF(%&gammaGF( F(F(\"\"!" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 258 "" 0 "" {TEXT 337 10 "Sous cas 1" }}{EXCHG {PARA 259 "" 0 "" {TEXT -1 3 " Si " }{XPPEDIT 18 0 "beta = 0;" "6#/%%betaG\"\"!" }{TEXT -1 12 ", alor s C : " }{TEXT 355 7 "X = - a" }{TEXT -1 15 " est une droite" }}}} {SECT 0 {PARA 258 "" 0 "" {TEXT 338 10 "Sous cas 2" }{TEXT -1 0 "" }} {EXCHG {PARA 259 "" 0 "" {TEXT -1 3 "Si " }{XPPEDIT 18 0 "beta <> 0;" "6#0%%betaG\"\"!" }{TEXT -1 12 ", alors C : " }{XPPEDIT 18 0 "X^2 = 2* p*Y;" "6#/*$%\"XG\"\"#*(F&\"\"\"%\"pGF(%\"YGF(" }{TEXT -1 21 " dans un rep\350re (O', " }{XPPEDIT 18 0 "e[1],e[2];" "6$&%\"eG6#\"\"\"&F$6#\" \"#" }{TEXT -1 43 " ). C est une parabole et on a B\262 - AC = 0." }}} }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 261 21 "Proc\351dure Quadratique" }{TEXT 268 1 "\n" } {TEXT -1 15 "Une expression " }{TEXT 262 5 "expr " }{TEXT -1 46 "est u ne forme quadratique en les variables X=[" }{XPPEDIT 18 0 "x[1],x[2]; " "6$&%\"xG6#\"\"\"&F$6#\"\"#" }{TEXT -1 3 "..." }{XPPEDIT 18 0 "x[n]; " "6#&%\"xG6#%\"nG" }{TEXT -1 37 "] si les 3 conditions sont v\351rifi \351es:" }}{PARA 0 "" 0 "" {TEXT 263 4 "expr" }{TEXT -1 30 " vaut 0 au point [0,0,...,0]. " }}{PARA 0 "" 0 "" {TEXT -1 10 "Pour tout " } {TEXT 264 1 "i" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "diff(expr,x[i]);" "6# -%%diffG6$%%exprG&%\"xG6#%\"iG" }{TEXT -1 68 " vaut 0 au point [0,0,.. .,0], c'est \340 dire que la diff\351rentielle de " }{TEXT 266 4 "expr " }{TEXT -1 32 " est nulle au point [0,0,...,0]." }}{PARA 0 "" 0 "" {TEXT -1 11 "Pour tout (" }{TEXT 265 3 "i,j" }{TEXT -1 4 "), " } {XPPEDIT 18 0 "diff(expr,x[j],x[i]);" "6#-%%diffG6%%%exprG&%\"xG6#%\"j G&F(6#%\"iG" }{TEXT -1 19 " est une constante." }{TEXT 267 1 "\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 379 "Quadratique:=proc(expr,X)\n local d,d1,d2,i,j;\nd:=subs(seq(X[i]=0,i=1..nops(X)),expr); \nif d<>0 \+ then return false fi;\nfor j to nops(X) do\n d1:=diff(expr,X[j]):\n \+ d:=subs(seq(X[i]=0,i=1..nops(X)),d1); \n if d<>0 then return false fi ;\nod;\nfor i to nops(X) do\n for j to nops(X) do\n d2:=diff(expr,X[ i],X[j]):\n if not type(d2,constant) then return false fi;\n od\nod; \ntrue\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "q:=5 *x^2+5*y^2+6*x*y: Quadratique(q,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 27 "Proc\351dure Ex traireFormeQuad" }{TEXT -1 66 "\nProc\351dure d'extraction de la parti e quadratique d'une expression " }{TEXT 270 4 "expr" }{TEXT -1 4 " en \+ " }{TEXT 271 1 "X" }{TEXT -1 2 ":\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "ExtraireFormeQuad:=proc(expr,X)\nlocal e,q;\nq:=0:\n for e in expr do\n if Quadratique(e,X) then q:=q+e fi\nend do: \nq\ne nd proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f:=5*x^2+5*y^ 2+6*x*y-4*x+4*y;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*&\"\"&\" \"\")%\"xG\"\"#F(F(*&F'F()%\"yGF+F(F(*(\"\"'F(F*F(F.F(F(*&\"\"%F(F*F(! \"\"*&F2F(F.F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "q:=Extr aireFormeQuad(f,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"qG,(*& \"\"&\"\"\")%\"xG\"\"#F(F(*&F'F()%\"yGF+F(F(*(\"\"'F(F*F(F.F(F(" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 272 44 "Proc\351dure de calcul du centre \+ de la conique\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 293 "centre: =proc(expr,X)\nlocal k,sys,c,s; \nsys:=\{seq(diff(expr,X[i]),i=1..nops (X))\};\ns:=solve(sys,convert(X,set)):\nc:=array(1..nops(X)):\nfor k t o nops(s) do\n if lhs(s[k])=X[1] then c[1]:=rhs(s[k])\n elif lhs(s[k ])=X[2] then c[2]:=rhs(s[k])\n else c[3]:=rhs(s[k]) fi\nod:\nconvert( c,list)\nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "cen tre(f,[x,y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7$\"\"\"!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 273 27 "Proc\351dure nouvelle_equation" } {TEXT 274 0 "" }{TEXT 275 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Cette pr oc\351dure effectue un changement de rep\350re: connaissant l'\351quat ion " }{TEXT 276 1 "f" }{TEXT -1 43 " dans l'ancien rep\350re, la nouv elle origine " }{TEXT 277 2 "pt" }{TEXT -1 24 ",\nla matrice de passag e " }{TEXT 278 1 "P" }{TEXT -1 52 " de l'ancienne \340 la nouvelle bas e, et les variables " }{TEXT 279 3 "var" }{TEXT -1 52 ", elle retourne l'\351quation dans le nouveau rep\350re.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 246 "nouvelle_equation:=proc(f,pt,P,var)\nlocal k,X, Y,Z,x,y,nx,s;\nnx:=[X,Y,Z]: \nx:=matrix([seq([nx[i]],i=1..nops(var))]) ;\ny:=evalm(P&*x); \ns:=seq(var[k]=y[k,1]+pt[k],k=1..nops(var)); unapp ly(simplify(expand(subs(s,f))),op(1..nops(var),nx)); \nend proc:\n" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "nouvelle_equation(f,[1,-1] ,matrix([[1/2*2^(1/2), 1/2*2^(1/2)], [-1/2*2^(1/2), 1/2*2^(1/2)]]),[x, y]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#f*6$%\"XG%\"YG6\"6$%)operatorG %&arrowGF',(*&\"\"#\"\"\")9$F-F.F.*&\"\")F.)9%F-F.F.\"\"%!\"\"F'F'F'" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 280 15 "Proc\351dure coord" }{TEXT 257 6 "\ncoord" }{TEXT -1 101 " donne les valeurs num\351riques des co ordonn\351es dans l'ancien rep\350re (o,i,j) d'un point de coordonn \351es " }{TEXT 258 3 "var" }{TEXT -1 16 " dans le nouveau" }}{PARA 0 "" 0 "" {TEXT -1 8 "rep\350re (" }{TEXT 259 2 "pt" }{TEXT -1 7 ",I,J). " }{TEXT 260 1 "P" }{TEXT -1 141 " est la matrice de passage de l'anc ienne \340 la nouvelle base. \nCette proc\351dure servira pour tracer \+ la conique dans le rep\350re initial (o,i,j).\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 143 "coord:=proc(var,pt,P)\nlocal x,y;\nx:=matrix( [seq([var[i]],i=1..nops(var))]);\ny:=evalm(P&*x);\nseq(evalf(y[k,1]+pt [k]),k=1..nops(var)) \nend proc:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "coord([sqrt(2),0],[1,-1],matrix([[1/2*2^(1/2), 1/2*2^ (1/2)], [-1/2*2^(1/2), 1/2*2^(1/2)]]));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$\"\"#\"\"!$!\"#F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 281 18 "Proc \351dure parabole" }{TEXT 282 0 "" }{TEXT 283 0 "" }{TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 116 "Cette proc\351dure traite le cas det(D)= 0 ( 0 est valeur propre) qui va donner la parabole comme seule conique propre.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1610 "parabole:=p roc(eq,P,xmin,xmax,ymin,ymax)\nlocal a,b,c,delta,e,s;\ne:=eq:\nif coef f(e,X,2)<>0 then e:=simplify(e/coeff(e,X,2)) \nelif coeff(e,Y,2)<>0 th en e:=simplify(e/coeff(e,Y,2)) fi:\nc:=subs(X=0,Y=0,e):\nif coeff(e,X, 2)=1 then\n a:=coeff(e,X,1): b:=coeff(e,Y,1): \n if b=0 then print(` Conique d\351g\351n\351r\351e:`):\n delta:=a^2-4*c:\n if delta <0 then print(`Ensemble vide`)\n elif delta=0 then print(`Droite d '\351quation:`):print(X=-a/2)\n else \n print(`r\351union des 2 droites d'\351quation:`): print( X=simplify((-a-sqrt(delta))/2),X=simplify((-a+sqrt(delta))/2))\n f i;\n else\n e:=(X+a/2)^2+b*(Y+(c-a^2/4)/b):\n print(`Parabole d'\351quation:`):print(e=0):\n s:=[-a/2,-(c-a^2/4)/b]:\n prin t(`de sommet: s`=s); \n print(`Equation dans (s,I,J):`); \n e :=X^2+b*Y : print(e=0): return elements_car_parabole(e,s,P,xmin,xmax,y min,ymax)\n fi\nelse\nif coeff(e,Y,2)=1 then\n a:=coeff(e,Y,1): b:=c oeff(e,X,1): \n if b=0 then print(`Conique d\351g\351n\351r\351e:`): \n delta:=a^2-4*c:\n if delta<0 then print(`Ensemble vide`)\n \+ elif delta=0 then print(`Droite d'\351quation:`):print(Y=-a/2)\n \+ else \n print(`r\351union des 2 droites d'\351quation:`): \+ print(Y=simplify((-a-sqrt(delta ))/2),Y=simplify((-a+sqrt(delta))/2))\n fi;\n else\n e:=(Y+a/ 2)^2+b*(X+(c-a^2/4)/b):\n print(`Parabole d'\351quation:`):print(e =0):\n s:=[-(c-a^2/4)/b,-a/2]:\n print(`de sommet: s`=s);\n \+ print(`Equation dans (s,I,J):`); \n e:=Y^2+b*X: print(e=0): retu rn elements_car_parabole(e,s,P,xmin,xmax,ymin,ymax) \n fi\nfi\nfi; \ne=0\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 284 31 "Proc\351dure el ements_car_parabole" }{TEXT 285 0 "" }{TEXT 286 0 "" }}{PARA 0 "" 0 " " {TEXT -1 69 "Cette proc\351dure donne les \351l\351ments caract\351r istiques de la parabole: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1128 "elements_car_parabole:=proc(eq,pt,P,xmin,xmax,ymin,ymax)\nlocal \+ p,axe1,axe2,courbe,points,dir;\nprint(`Elements caract\351ristiques da ns (s,I,J):`);\nif coeff(eq,Y,2)=1 then p:=simplify(-1/2*coeff(eq,X,1) ):print(`axe focal: `=(s,I))\n else p:=simplify(-1/2*coeff(eq,Y,1)):p rint(`axe focal: `=(s,J)) fi:\naxe1:=plot([pt[1]+t*P[1,1],pt[2]+t*P[2, 1],t=-15..15],linestyle=DOT):\naxe2:=plot([pt[1]+t*P[1,2],pt[2]+t*P[2, 2],t=-15..15],linestyle=DOT):\nprint(`param\350tre: p`=p);\nprint(`ex centricit\351: e`=1);\npoints:=pt;\nprint(`foyer:`);\nif coeff(eq,Y,2 )=1 then print(F(p/2,0)):points:=points,[coord([p/2,0],pt,P)]\nelse pr int(F(0,p/2)):points:=points,[coord([0,p/2],pt,P)] fi:\npoints:=pointp lot([points],symbol=CROSS);\nprint(`directrice:`);\nif coeff(eq,Y,2)=1 then print(X=-p/2):dir:=plot([coord([-p/2,t],pt,P),t=-15..15],color=b lue) else print(Y=-p/2):dir:=plot([coord([t,-p/2],pt,P),t=-15..15],col or=blue) fi:\nif coeff(eq,Y,2)=1 then courbe:=plot([coord([t^2/(2*p),t ],pt,P),t=-15..15]) else\ncourbe:=plot([coord([t,t^2/(2*p)],pt,P),t=-1 5..15]) fi;\ndisplay(\{courbe,axe1,axe2,points,dir\},scaling=constrain ed,view=[xmin..xmax,ymin..ymax])\nend proc:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 287 30 "Proc\351dure elements_car_ellipse" }{TEXT 288 0 "" } {TEXT 289 0 "" }}{PARA 0 "" 0 "" {TEXT -1 67 "Cette proc\351dure donne les \351l\351ments caract\351ristiques de l'ellipse: \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1836 "elements_car_ellipse:=proc(a,b,pt ,P,xmin,xmax,ymin,ymax)\nlocal c1,e,axe1,axe2,courbe,points,dirs;\npri nt(`Elements caract\351ristiques dans (c,I,J):`);\nif evalf(a)<>evalf( b) then\n if evalf(a)>evalf(b) then c1:=sqrt(a^2-b^2): e:=c1/a \n else c1:=sqrt(b^2-a^2): e:=c1/b fi:\n print(` a`=simplify(a),` b`=simplify (b), ` c`=simplify(c1));\n if evalf(a)>evalf(b) then print(`grand axe \+ (ou axe focal): `=(c,I))\n else print(`grand axe (ou axe focal): `=(c, J)) fi:\nfi;\naxe1:=plot([pt[1]+t*P[1,1],pt[2]+t*P[2,1],t=-15..15],lin estyle=DOT):\naxe2:=plot([pt[1]+t*P[1,2],pt[2]+t*P[2,2],t=-15..15],lin estyle=DOT):\nif evalf(a)<>evalf(b) then\n print(`excentricit\351 (0< e<1) e`=simplify(e));\n print(`sommets:`);\n print(A1(-a,0),A2(a,0) ,B1(0,-b),B2(0,b));\n print(`foyers:`);\n if evalf(a)>evalf(b) then \+ print(F1(-c1,0),F2(c1,0)) else print(F1(0,-c1),F2(0,c1)) fi;\n points :=[coord([-a,0],pt,P)],[coord([a,0],pt,P)],[coord([0,-b],pt,P)],[coord ([0,b],pt,P)]:\n if evalf(a)>evalf(b) then points:=points,[coord([-c1 ,0],pt,P)],[coord([c1,0],pt,P)]\n else points:=points,[coord([0,-c1], pt,P)],[coord([0,c1],pt,P)] fi:\n points:=pointplot([points],symbol=C ROSS);\n print(`directrices:`);\n if evalf(a)>evalf(b) then\n pri nt(X=simplify(-a^2/c1),X=simplify(a^2/c1)): dirs:=plot([coord([- a^2/c1,t],pt,P),t=-15..15],color=blue),plot([coord([a^2/c1,t],pt,P),t= -15..15],color=blue)\n else print(simplify(Y=-b^2/c1),simplify(X=b^2/ c1)):\ndirs:=plot([coord([t,-b^2/c1],pt,P),t=-15..15],color=blue),plot ([coord([t,b^2/c1],pt,P),t=-15..15],color=blue):\n fi;\nfi:\ncourbe:= plot([coord([a*cos(t),b*sin(t)],pt,P),t=0..2*Pi]);\nif evalf(a)<>evalf (b) then\n display(\{courbe,axe1,axe2,points,dirs\},scaling=constrain ed,view=[xmin..xmax,ymin..ymax])\nelse display(courbe,pointplot([[coor d([0,0],pt,P)]],symbol=CROSS),scaling=constrained,view=[xmin..xmax,ymi n..ymax])\nfi\nend proc:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 290 32 "Pr oc\351dure elements_car_hyperbole" }{TEXT 291 0 "" }{TEXT 292 0 "" }} {PARA 0 "" 0 "" {TEXT -1 69 "Cette proc\351dure donne les \351l\351men ts caract\351ristiques de l'hyperbole: \n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1892 "elements_car_hyperbole:=proc(a,b,s,pt,P,xmin,xmax ,ymin,ymax)\nlocal e,c1,axe1,axe2,courbe,points,dirs,asys;\nprint(`Ele ments caract\351ristiques dans (c,I,J):`);\nc1:=sqrt(a^2+b^2):\nif s=1 then e:=c1/a else e:=c1/b fi:\nprint(` a`=simplify(a),` b`=simplify(b ), ` c`=simplify(c1));\nif s=1 then print(`axe focal (ou transverse): \+ `=(c,I))\nelse print(`axe focal (ou transverse): `=(c,J)) fi:\naxe1:=p lot([pt[1]+t*P[1,1],pt[2]+t*P[2,1],t=-15..15],linestyle=DOT):\naxe2:=p lot([pt[1]+t*P[1,2],pt[2]+t*P[2,2],t=-15..15],linestyle=DOT):\nprint(` excentricit\351 (e>1) e`=simplify(e));\nprint(`sommets:`);\nif s=1 th en print(A1(-a,0),A2(a,0)):points:=[coord([-a,0],pt,P)],[coord([a,0],p t,P)]\nelse print(B1(0,-b),B2(0,b)): points:=[coord([0,-b],pt,P)],[coo rd([0,b],pt,P)]: fi;\nprint(`foyers:`);\nif s=1 then print(F1(-c1,0),F 2(c1,0)):points:=points,[coord([-c1,0],pt,P)],[coord([c1,0],pt,P)]\nel se print(F1(0,-c1),F2(0,c1)):points:=points,[coord([0,-c1],pt,P)],[coo rd([0,c1],pt,P)] fi;\npoints:=pointplot([points],symbol=CROSS);\nprint (`directrices:`);\nif s=1 then print(X=simplify(-a^2/c1),X=simplify(a^ 2/c1)):\ndirs:=plot([coord([-a^2/c1,t],pt,P),t=-15..15],color=blue),pl ot([coord([a^2/c1,t],pt,P),t=-15..15],color=blue) else print(simplify( Y=-b^2/c1),simplify(X=b^2/c1)):\ndirs:=plot([coord([t,-b^2/c1],pt,P),t =-15..15],color=blue),plot([coord([t,b^2/c1],pt,P),t=-15..15],color=bl ue): fi;\nprint(`asymptotes:`);\nprint(Y=-b/a*X,Y=b/a*X);\nasys:=plot( [coord([t,-b/a*t],pt,P),t=-15..15],color=black,linestyle=DOT),plot([co ord([t,b/a*t],pt,P),t=-15..15],color=black,linestyle=DOT):\nif s=1 the n \ncourbe:=plot([coord([a*cosh(t),b*sinh(t)],pt,P),t=-5..5]),plot([co ord([-a*cosh(t),b*sinh(t)],pt,P),t=-5..5])\nelse courbe:=plot([coord([ a*sinh(t),b*cosh(t)],pt,P),t=-5..5]),plot([coord([a*sinh(t),-b*cosh(t) ],pt,P),t=-5..5]) fi;\ndisplay(\{courbe,axe1,axe2,points,dirs,asys\},s caling=constrained,view=[xmin..xmax,ymin..ymax])\nend proc:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 293 17 "Proc\351dure conique" }{TEXT 294 0 "" }{TEXT 295 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "C'e st la proc\351dure principale de la feuille. Elle re\347oit en param \350tres: " }}{PARA 0 "" 0 "" {TEXT 307 2 "- " }{TEXT -1 16 " une expr ession " }{TEXT 296 5 "expr " }{TEXT -1 14 "des variables " }{TEXT 297 3 "var" }{TEXT -1 2 ".\n" }{TEXT 308 1 "-" }{TEXT -1 14 " quatre r \351els " }{TEXT 298 4 "xmin" }{TEXT -1 1 "," }{TEXT 300 5 " xmax" } {TEXT -1 2 ", " }{TEXT 301 4 "ymin" }{TEXT -1 1 "," }{TEXT 302 5 " yma x" }{TEXT -1 1 "." }{TEXT 304 3 " \n " }{TEXT -1 34 "Elle extrait la f orme quadratique " }{TEXT 303 1 "q" }{TEXT -1 12 " associ\351e \340 " }{TEXT 305 4 "expr" }{TEXT -1 81 ", r\351duit sa matrice, calcule une \+ base [I,J] de IR\262 orthonormale et qui est aussi " }{TEXT 306 1 "q" }{TEXT -1 140 "-orthogonale,\nanalyse la nature de la conique et donne ses \351l\351ments caract\351ristiques ainsi que sa repr\351sentation graphique dans la fen\352tre [" }{TEXT 309 4 "xmin" }{TEXT -1 1 "," } {TEXT 310 5 " xmax" }{TEXT -1 2 ", " }{TEXT 311 4 "ymin" }{TEXT -1 1 " ," }{TEXT 312 5 " ymax" }{TEXT -1 4 "].\n " }{TEXT 299 1 " " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2439 "conique:=proc(expr,var,xmi n,xmax,ymin,ymax)\nlocal e,k,q,v,A,D,G,P,c,eq,sm,vp,alpha,beta,gamma,a ,b;\nq:=ExtraireFormeQuad(expr,var);\nfor k in expr-q do\n if not typ e(k,constant) then\n for v in var do \n if not type(diff(k,v), constant) then error \n\"Cette expression ne correspond pas \340 la fo rme attendue: Ax\262+2Bxy+Cy\262+Dx+Ey+F\" fi\n od\n fi\nod;\nif q =0 then error \"Forme quadratique associ\351e nulle\" fi:\nprint(`Form e quadratique associ\351e: q`=q);\nprint(`Matrice de la forme quadrat ique associ\351e:`);\nA:=matrix([seq([seq(diff(q,var[i],var[j])/2,j=1. .2)],i=1..2)]);\nprint(`A `=evalm(A));\nvp:=sort([eigenvals(A)]):\npri nt(`Valeur propres de A `=vp);\nD:=jordan(A,'P1');\nprint(`Matrice dia gonale semblable \340 A: D`=evalm(D));\nG:=map(normalize, GramSchmid t([col(P1,1..2)])); \nP:=map(simplify,concat(op(G)));\nprint(`Matrice \+ de passage orthogonale: P`=evalm(P));\nprint(`Directions principales de la conique:`);\nprint(`I `=col(P,1),` J `=col(P,2));\nif det(D)<>0 then \n c:=centre(expr,var): print(`Conique \340 centre: c`=c): \n print(`Equation dans (c,I,J): `):\n eq:=nouvelle_equation(f,c,P,var ):\n alpha:=-eq(0,0);\n print(collect(eq(X,Y),[X,Y])=0): \n if alp ha=0 then\n print(`Conique d\351g\351n\351r\351e`);\n if det(D)> 0 then \n print(`r\351duite au centre: c`=\{c\}) \n else pri nt(`r\351union des 2 droites d'\351quation:`);\n print(Y=sqrt(-vp [1]/vp[2])*X, Y=-sqrt(-vp[1]/vp[2])*X) \n fi\n else # alpha<>0\n \+ if evalf(det(D))>0 then\n if evalf(alpha/vp[1])<0 and eval f(alpha/vp[2])<0 then print(eq(X,Y)=0): \n print(`Coniqu e d\351g\351n\351r\351e: ensemble vide`) \n else a:=sqrt(alpha/vp [1]): b:=sqrt(alpha/vp[2]):\n print(X^2/a^2+Y^2/b^2=1): \+ \n if evalf(a)<>evalf(b) then print(`Ellipse de centre c`) el se \n print(`cercle de centre c, rayon`=a) fi:\n elem ents_car_ellipse(a,b,c,P,xmin,xmax,ymin,ymax):\n fi\n else # d et(D)<0\n if evalf(alpha/vp[1])>0 and evalf(alpha/vp[2])<0 then \+ \n a:=sqrt(alpha/vp[1]): b:=sqrt(-alpha/vp[2]): sm:=1:\n \+ print(X^2/a^2-Y^2/b^2=1):\n else\n a:=sqrt(-alpha/vp[1]) : b:=sqrt(alpha/vp[2]): sm:=-1:\n print(X^2/a^2-Y^2/b^2=-1):\n fi:\n print(`Hyperbole de centre c`):\n elements_car_h yperbole(a,b,sm,c,P,xmin,xmax,ymin,ymax):\n fi \n fi\nelse # de t(D)=0\n print(`Equation dans (o,I,J): `):\n eq:=nouvelle_equation(f ,[0,0],P,var):\n print(eq(X,Y)=0);\n parabole(eq(X,Y),P,xmin,xmax,ym in,ymax);\n fi:\nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 313 43 "Exem ple 1 : Affichage d'un message d'erreur" }{TEXT 314 0 "" }{TEXT 315 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f:=5*x^2+5*y^2+6*x*y -4*x+4*y-12*x^2*y^2; conique(f,[x,y],-10,10,-10,10);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,.*&\"\"&\"\"\")%\"xG\"\"#F(F(*&F'F()%\"yGF+F( F(*(\"\"'F(F*F(F.F(F(*&\"\"%F(F*F(!\"\"*&F2F(F.F(F(*(\"#7F(F)F(F-F(F3 " }}{PARA 8 "" 1 "" {TEXT -1 97 "Error, (in conique) Cette expression \+ ne correspond pas \340 la forme attendue: Ax\262+2Bxy+Cy\262+Dx+Ey+F\n " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 316 31 "Exemple 2 : Etude d'une elli pse" }{TEXT 317 0 "" }{TEXT 318 1 "\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "f:=5*x^2+5*y^2+6*x*y-4*x+4*y; conique(f,[x,y],-1,3,-3 ,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fG,,*&\"\"&\"\"\")%\"xG\" \"#F(F(*&F'F()%\"yGF+F(F(*(\"\"'F(F*F(F.F(F(*&\"\"%F(F*F(!\"\"*&F2F(F. F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%?Forme~quadratique~associ|dy e:~~qG,(*&\"\"&\"\"\")%\"xG\"\"#F(F(*&F'F()%\"yGF+F(F(*(\"\"'F(F*F(F.F (F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%JMatrice~de~la~forme~quadratiq ue~associ|dye:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#A~GK%'matrixG6#7 $7$\"\"&\"\"$7$F+F*Q(pprint06\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%5 Valeur~propres~de~A~G7$\"\"#\"\")" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/ %EMatrice~diagonale~semblable~|[y~A:~~~DGK%'matrixG6#7$7$\"\"#\"\"!7$F +\"\")Q(pprint16\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%DMatrice~de~pa ssage~orthogonale:~~~PGK%'matrixG6#7$7$,$*&\"\"#!\"\"F,#\"\"\"F,F/F*7$ ,$*&F,F-F,F.F-F*Q(pprint26\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%FDire ctions~principales~de~la~conique:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$ /%#I~GK%'vectorG6#7$,$*&\"\"#!\"\"F+#\"\"\"F+F.,$*&F+F,F+F-F,Q(pprint3 6\"/%$~J~GKF&6#7$F)F)Q(pprint4F2" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/% 6Conique~|[y~centre:~~~cG7$\"\"\"!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%8Equation~dans~(c,I,J):~G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/, (*&\"\"#\"\"\")%\"XGF&F'F'*&\"\")F')%\"YGF&F'F'\"\"%!\"\"\"\"!" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"#!\"\"%\"XGF&\"\"\"*&F&F))%\" YGF&F)F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%4Ellipse~de~centre~cG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%HElements~caract|dyristiques~dans~( c,I,J):G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6%/%#~aG*$\"\"##\"\"\"F&/%#~ bG,$*&F&!\"\"F&F'F(/%#~cG,$*&F&F-\"\"'F'F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%;grand~axe~(ou~axe~focal):~G6$%\"cG^#\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%8excentricit|dy~(0X5F*7$$!3\"GuY(y9*)H5F*$!3sFuY 6P^*)**!#<7$$!3_!))zg#Rf'*)*F2$!3?!))z+&G>(e*F27$$!3'oAC+/UTW*F2$!3`EU -k4uM\"*F27$$!3k^0<45p))*)F2$!3I^0Z'F27$$!3aH1Od>&*>jF2$!3?H1O\")3b5gF27$$!3YZw'RWDN\"fF2$!37Zw'zOCT g&F27$$!3!)Q]u^a)fX&F2$!3YQ]uvVeY^F27$$!3QE[Ncnc'*\\F2$!31E[N!olro%F27 $$!3i@A8:<$Qb%F2$!3I@A8R1VWUF27$$!3CVr#H)ey^TF2$!3MVr#p![QUQF27$$!3-i) 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