restart:with(plots):

Warning, the name changecoords has been redefined

EXERCICE 5

racines:=proc(P,X) #P est une expression polynomiale de degr\351 2 en X local a,b,c,d,r1,r2; c:=subs(X=0,P):b:=subs(X=0,diff(P,X)):a:=subs(X=0,diff(P,X$2))/2: #P(X)=aX^2+bX+c d:=b^2-4*a*c: #d est le discriminant de P if is(d>=0) then r1:=(-b-sqrt(d))/(2*a): r2:=(-b+sqrt(d))/(2*a); else r1:=(-b- I*sqrt(-d))/(2*a): r2:=(-b+I*sqrt(-d))/(2*a); fi; if is(a>0) then [r1,r2]; else [r2,r1]; fi; end:

a:=5:b:=3:F:=(x,y)->x^2/a^2+y^2/b^2-1;

NiM+SSJGRzYiZio2JEkieEdGJUkieUdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCgqJjkkIiIjSSJhR0YlISIjIiIiKiY5JUYwSSJiR0YlRjJGMyEiIkYzRiVGJUYl

vec_normal:=proc(F,M) #fonction d\351finissant la conique et M point o\371 l'on calcule la normale local u,n,norme; u:=[diff(F(x,y),x),diff(F(x,y),y)]; n:=subs(x=M[1],y=M[2],u); norme:=sqrt(n[1]^2+n[2]^2): #Calcule la norme du vecteur n n:=[n[1]/norme,n[2]/norme]: #normalise le vecteur end:

impact_ellipse:=proc(M,d) local r,t,n,ps,P,delta; r:=racines(F(M[1]+t*d[1],M[2]+t*d[2]),t); P:=[M[1]+r[2]*d[1],M[2]+r[2]*d[2]]; n:=vec_normal(F,P):ps:=d[1]*n[1]+d[2]*n[2]: delta:=[d[1]-2*ps*n[1],d[2]-2*ps*n[2]]; [P,delta]; end:

P:=[0,0];theta:=Pi/4;

NiM+SSJQRzYiNyQiIiFGJw==NiM+SSZ0aGV0YUc2IiwkSSNQaUdJKnByb3RlY3RlZEdGKCMiIiIiIiU=

d:=[cos(theta),sin(theta)]:trajet:=P:#d:=[4/5,3/5];for i from 1 to 5 do Imp:=impact_ellipse(P,d); P:=Imp[1]:d:=Imp[2]:trajet:=trajet,P; od:trajet;

Nig3JCIiIUYkNyQsJComIiM8IyIiIiIiI0YrRikjIiM6IiNNRiY3JCwkRicjIihsM0IkIigxXj4nLCRGJyMhKCY0SURGMzckLCRGJyMhMmw/N1NSbCNRZSIzJz1UVVokMzZtPCwkRicjIjJ2QHZQMzNVUSlGOzckLCRGJyMhQjp3MDQmZnNLKik0RXI7RyhHMyciQyU+U1Q0OmkuW3VpI1soeXkvMSIsJEYnIyFCJnByYS0hKlxgeHgxZnBecWJTRkM3JCwkRicjIlc6Zzx4ZClwaScpKVFaTDoyNGckWydvYXYsbSgpeSJvIiJXTTsvSThtJUckKXpNaC5cPDtlJSlRQkJ0Xm1EUXYpLCRGJyMiV04iNFQpW3ArMDZaZktScCZlYXdPTWRUQiYpNCQqUSVGSw==

Boule:=plot([trajet],style=line,thickness=2):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue):display({Boule, Billard});

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

b:=3:P:=[-5,0];

NiM+SSJQRzYiNyQhIiYiIiE=

d:=[5/sqrt(34),3/sqrt(34)];trajet:=P:

NiM+SSJkRzYiNyQsJCokIiNNIyIiIiIiIyMiIiZGKSwkRigjIiIkRik=

for i from 1 to 150 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:

Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):display({Boule, Billard},axes=none);

-%%PLOTG6'-%'CURVESG6&7ct7$$!"&""!$F,F,7$F-$""$F,7$$"3#*******)*******\!#<$!31+++++++5!#E7$$!3'**************R"!#D$!33+++,+++IF47$$!3e*****\*******\F4$"3)***************HF77$$"31+++++++8F;$"33+++,+++IF47$F2$!35+++++++]F77$$!3#**************z#F;$!3#*******)*******HF47$$!3L+++/+++]F4$"3#**************4"F;7$$"33+++++++uF;$"3%)*****z*******HF47$$""&F,$!37+++++++>F;7$$!3-++++++]5!#CFN7$$!3w*****p*******\F4$"3-+++++++FF;7$$"37++++++g9F\oFF7$Fen$!33+++++++SF;7$$!3!**************=#F\o$!3%)*****z*******HF47$F^o$"3++++++++^F;7$$"3;++++++gDF\o$"3#*******)*******HF47$F2$!3))**************eF;7$$!3"*************HHF\oF<7$$!33+++,+++]F4$"3W+++++++nF;7$$"3#)************fKF\oFcp7$$"3%)*****z*******\F4$!3[+++++++qF;7$$!3!*************RKF\oFN7$F\q$"3_+++++++tF;7$$"3)*************HNF\oFcp7$$"3;+++-+++]F4$!3e+++++++zF;7$$!3)*************RQF\oF<7$F*$"3#)*************p)F;7$$"3!)*************=%F\oF/7$$"3T+++0+++]F4$!3)***************)*F;7$$!3`++++++5]F\oF<7$$!3;+++-+++]F4$"3)*************\6F\o7$$"3#*************\cF\oFF7$Fen$!35++++++!G"F\o7$$!3E++++++!>'F\oF<7$$!3#*******)*******\F4$"30++++++59F\o7$$"31++++++]oF\oFcp7$$"3w*****p*******\F4$!32++++++I:F\o7$$!3V++++++?tF\o$!"$F,7$FQ$"3'*************z;F\o7$$"3%)*************>)F\oFcp7$Fdq$!3*)************>=F\o7$$!3A++++++q')F\oFeu7$Fhs$"3"*************R>F\o7$$"3#**************H*F\oF/7$$"3m*****f*******\F4$!30++++++I?F\o7$$!3*4++++++a*F\oF<7$Fft$"3*)************H@F\o7$$"3-++++++:5!#BFX7$Fen$!35++++++qAF\o7$$!3'**************3"FgwF<7$F\q$"31++++++gCF\o7$$"3)*************)="FgwFF7$$"33+++,+++]F4$!31++++++]EF\o7$$!3'************\E"FgwF<7$$!3m*****f*******\F4$"3)**************y#F\o7$$"3)************fJ"FgwF/7$FenFip7$$!31++++++'R"FgwFN7$Fft$"3>++++++5JF\o7$$"31++++++'["Fgw$"3;+++-+++IF47$Fbr$!3+++++++SLF\o7$$!33++++++-;FgwF<7$FftF_r7$$"3'************zm"FgwF^z7$F2$!39++++++!o$F\o7$$!3&*************H<FgwF[p7$F\q$"3%**************y$F\o7$$"3%************fy"Fgw$"3*)*****R++++$F47$Fbr$!3<++++++?RF\o7$$!3%************R$=FgwFN7$F*$"3#)**************RF\o7$$"3'************z'=FgwF/7$Fbr$!3C++++++!4%F\o7$$!3$)************>>FgwFN7$$!3D+++.+++]F4F]s7$$"3/++++++i>FgwFcp7$Fbr$!3+++++++qUF\o7$$!3;++++++#*>FgwF<7$F\q$"3F++++++IVF\o7$$"3*************z,#Fgw$"3!)*****H++++$F47$$"3D+++.+++]F4$!3y************zVF\o7$$F]wFgwFeu7$F[]l$"3y************zVF\o7$$"3&************H-#FgwFcp7$F2$!3#)************4VF\o7$$!31++++++o>FgwFeu7$F\q$"3%*************>UF\o7$$"31++++++S>FgwF^z7$Fbr$!3)*************\TF\o7$$!3-+++++++>FgwFeu7$Fft$"3'*************HSF\o7$$"3)*************G=FgwF/7$$"3e*****\*******\F4$!3w************HQF\o7$$!3!************zr"Fgw$!3;+++-+++IF47$Fft$"3<++++++5OF\o7$$"35++++++A;FgwF/7$Fen$!3))************HMF\o7$$!3+++++++`:FgwFeu7$$!3%)*****z*******\F4$"37++++++]KF\o7$$"3#*************e9FgwF^z7$Fen$!3#*************\IF\o7$$!3)************4O"FgwF[p7$F\q$"3C++++++]GF\o7$$"31++++++'G"FgwFF7$Fjv$!3))*************o#F\o7$$!3"************z?"FgwF<7$F[]l$"33++++++!e#F\o7$$"3#************p<"FgwF/7$Fen$!31++++++gCF\o7$$!3$************R5"Fgw$!3!)*****H++++$F47$Fjal$"3/++++++!G#F\o7$$"3U************p**F\oFX7$F`s$!37++++++!3#F\o7$$!3M++++++?$*F\oFN7$Fjal$"3**************>>F\o7$$"3K++++++g%)F\oFX7$F`s$!3))************f<F\o7$$!3)*************>zF\oF[al7$Fjal$"3/+++++++;F\o7$$"3L++++++5pF\oF^z7$Fen$!3!**************Q"F\o7$$!3g************pfF\oF[al7$F^oFjs7$$"3/++++++IYF\oF/7$Fdq$!3#)*************p)F;7$$!3'*************4MF\oFN7$Fjal$"3Y**************fF;7$$"31++++++]@F\oFcp7$F2$!3y*************H$F;7$$!3U*************4*F;F<7$F*$"3'***************fF77$$!3A+++++++NF;F/7$Fdq$"3%)*************>#F;7$$"31++++++g;F\oF<7$F*$!3;+++++++YF;7$$!38++++++IEF\oFcp7$Fbr$"3S+++++++kF;7$$"3')************4LF\oF[p7$Fft$!3i+++++++#)F;7$$!3')************RUF\oFcp7$Fex$"3)***************)*F;7$$"3I++++++!)[F\oF<7$Fjal$!3-++++++S6F\o7$$!3[+++++++dF\oFcp7$Fbr$"3&*************48F\o7$$"3+++++++qjF\oF<7$Fhs$!3$*************R9F\o7$$!3+++++++!*pF\oFF7$F_^l$"3.++++++S:F\o-%&STYLEG6#%%LINEG-%*THICKNESSG6#"""-%&COLORG6&%$RGBG$"#5!""$F,Fb\mFc\m-F&6ds7$7$$!3k*************f%F4$!3yLLLLLLf6F47$$!3")["["["[1r%F4$!2y***************F47$F\]m7$$!3+@'eF[Mg![F4$!3r)y8C<b'Rz!#=7$7$$!3-LLLLL$e*[F4$!3m)*************fFg]mFb]m7$Fi]m7$$!3%[9O!f(R\)\F4$!3K_&Q'4Cg]@Fg]m7$7$$!3gDf#f#fU))\F4$!3W)*************>Fg]mF_^m7$7$$!3[Ef#f#fU))\F4$!3s)*************>Fg]m7$Ff^m$"3Oh#f#f#fU)=Fg]m7$7$Ff^m$"3w,++++++?Fg]mF`_m7$7$F\_mFe_m7$$!3N'eMQ`8a"\F4$"3oneMQ`8a^Fg]m7$7$Fj]m$"3)>+++++++'Fg]mFi_m7$F_`m7$$!3*oTt)*=:A![F4$"3+!=M()*=:A!)Fg]m7$7$F]]m$"3A+++++++5F4Fc`m7$Fi`m7$$!3+(3[ka+`m%F4$"3!y3[ka+`1"F47$7$Fh\m$"3+MLLLLLf6F4F]am7$7$$!3G*************>%F4$!3#3+++++bh"F47$$!3knnnnnn<WF4$!3!**************R"F47$F\bm7$Fh\m$!3cLLLLLLf6F47$Fcam7$$!3S"4;yOki\%F4$"3K#4;yOkiH"F47$7$F]bm$"3M+++++++9F4Ffbm7$F\cm7$$!3[JkGd9H3VF4$"3bKkGd9H3:F47$7$Fham$"3#3+++++bh"F4F`cm7$7$$!3#*)************z$F4$!3M+++++SS>F47$$!35++++++&*RF4$!3#)*************z"F47$F_dmFgam7$Ffcm7$$!3r/@%ot%*G5%F4$"3"f5Uot%*Gq"F47$7$F`dm$"3E+++++++=F4Ffdm7$7$$!3m***********\*RF4F]em7$$!3UVn(p!zi")QF4$"3aWn(p!zi")=F47$7$$!3Q*************z$F4$"3M+++++SS>F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Voici d'autres rebonds possibles depuis des configurations initiales distinctes

b:=3:P:=[-5/3,0];

NiM+SSJQRzYiNyQjISImIiIkIiIh

d:=[5/sqrt(34),3/sqrt(34)];trajet:=P:

NiM+SSJkRzYiNyQsJCokIiNNIyIiIiIiIyMiIiZGKSwkRigjIiIkRik=

for i from 1 to 150 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:

Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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b:=5:P:=[1,0];theta:=Pi/2:

NiM+SSJQRzYiNyQiIiIiIiE=

d:=[cos(theta),sin(theta)];trajet:=P:

NiM+SSJkRzYiNyQiIiEiIiI=

for i from 1 to 100 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:trajet:Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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Question b: Billard CirculaireP=[-5,0]:theta=Pi/3:

d:=[cos(theta),sin(theta)];trajet:=P:

NiM+SSJkRzYiNyQiIiEiIiI=

for i from 1 to 100 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:trajet:Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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P=[-5,0]:theta=Pi/6:

d:=[cos(theta),sin(theta)];trajet:=P:

NiM+SSJkRzYiNyQiIiEiIiI=

for i from 1 to 100 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:trajet:Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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P=[-5,0]:theta=1:

d:=[cos(theta),sin(theta)];trajet:=P:

NiM+SSJkRzYiNyQiIiEiIiI=

for i from 1 to 100 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:trajet:Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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P=[-3,0]:theta=Pi/6:

d:=[cos(theta),sin(theta)];trajet:=P:

NiM+SSJkRzYiNyQiIiEiIiI=

for i from 1 to 100 do Imp:=impact_ellipse(P,d); P:=evalf(Imp[1]):d:=evalf(Imp[2]):trajet:=trajet,P; od:trajet:Boule:=plot([trajet],style=line,thickness=1):Billard:=implicitplot(F(x,y)=0,x=-5..5,y=-5..5,color=blue,thickness=1):

display({Boule, Billard},axes=none);

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