Exercice 2 On d\351fnit la param\351trisation, puis on construit le support partiel et sir la totalit\351 des param\350tresOn d\351fnit la param\351trisation, puis on construit le support partiel et sir la totalit\351 des param\350tres

restart;

x:=t->t/(t^2-1);y:=t->t^2/(t-1);

NiM+SSJ4RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIiwmKiRGLSIiI0YuISIiRi5GMkYlRiVGJQ==

NiM+SSJ5RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiY5JCIiIywmRi0iIiIhIiJGMEYxRiVGJUYl

total:=[x(t),y(t),t=-infinity..infinity]:

plot(total);

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

Cherchons le points multiples

1ere M\351thode

plot(x,-2..2,-10..10,discont=true);

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On voit que x est impaire et que tout point image admet deux ant\351c\351dents

tx:=solve(x(t)=u,t);

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V\351rifions si pour ces deux ant\351c\351dents t1 et t2 d'un point image u de x, les images par y sont les m\352mes

y1:=y(tx[1]);y2:=y(tx[2]);

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NiM+SSN5Mkc2IiwkKigsJiEiIiIiIiokLCZGKkYqKiRJInVHRiUiIiMiIiUjRipGL0YqRi9GLiEiIywmKiZGKEYqRi5GKSNGKUYvRilGKkYpI0YqRjA=

Calculons donc pour quelle valeur de u, les param\350tres t1 et t2 ont m\352mes images par y

solve(y1=y2,u);

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seule la valeur r\351elle de u, convient, i.e. u=-1. Et donc on a un point double dont les param\350tres sont

t_double:=solve(x(t)=-1,t);

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il s'agit du point D de coordonn\351es

simplify([x(t_double),y(t_double)]);

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Ainsi le point O de pam\351tre 0 est un point triple, les autres sont des points simples.

Tracons le support de la courbe autour de ce point double

plot(total,-5..5,-2..1);

Attention MAPLE semble d\351tecter d'autres points multiples... Pourquoi???

2eme M\351thode

Autre m\351thode: plut\364t que partir des ant\351c\351dents de x, prenons ceux de y

plot(y,-10..10,-10..10,discont=true);

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

l\340 encore, on voit que tout point image admet deux ant\351c\351dents

ty:=solve(y(t)=u,t);

NiM+SSN0eUc2IjYkLCZJInVHRiUjIiIiIiIjKiQsJiokRihGK0YqRighIiVGKUYpLCZGKEYpRiwjISIiRis=

V\351rifions si pour ces deux ant\351c\351dents t1 et t2 d'un point image u de y, les images par x sont les m\352mes

x1:=x(ty[1]);x2:=x(ty[2]);

NiM+SSN4MUc2IiomLCZJInVHRiUjIiIiIiIjKiQsJiokRihGK0YqRighIiVGKUYpRiosJiokRidGK0YqISIiRipGMg==

NiM+SSN4Mkc2IiomLCZJInVHRiUjIiIiIiIjKiQsJiokRihGK0YqRighIiVGKSMhIiJGK0YqLCYqJEYnRitGKkYxRipGMQ==

Calculons donc pour quelle valeur de u, les param\350tres t1 et t2 ont m\352mes image par x

solve(x1=x2,u);

NiUhIiIiIiEiIiU=

Cependant, ici il y a un pi\350ge, en effet si u vaut 0 ou 4 Alors t1=t2 (voir expressions de ty) et donc il ne s'agit plus d'un point double, car le param\350tre est le m\352me

il ne reste donc que le cas u=-1, c'est \340 dire le point d'ordonn\351e -1 et pour param\350tres

solve(y(t)=-1,t);

NiQsJiMhIiIiIiMiIiIqJCIiJiNGJ0YmRiosJkYkRidGKEYk

CQFD

3eme M\351thode

Derni\350re M\351thode (Le Meilleur pour la fin). Essayons sans trop de calcul de retrouver le r\351sultat

Soient t1 et t2 deux param\350tres distincts correspondants \340 un point double. Le couple (t1,t2) est un couple solution du syst\351me

systeme:=[x(t1)-x(t2)=0,y(t1)-y(t2)=0];

NiM+SShzeXN0ZW1lRzYiNyQvLCYqJkkjdDFHRiUiIiIsJiokRioiIiNGKyEiIkYrRi9GKyomSSN0MkdGJUYrLCYqJEYxRi5GK0YvRitGL0YvIiIhLywmKiZGKkYuLCZGKkYrRi9GK0YvRisqJkYxRi4sJkYxRitGL0YrRi9GL0Y0

On simplifie les expressions

systeme:=factor(systeme);

NiM+SShzeXN0ZW1lRzYiNyQvLCQqLiwmKiZJI3QyR0YlIiIiSSN0MUdGJUYtRi1GLUYtRi0sJkYsISIiRi5GLUYtLCZGLEYtRjBGLUYwLCZGLEYtRi1GLUYwLCZGLkYtRjBGLUYwLCZGLkYtRi1GLUYwRjAiIiEvKiosKEYrRi1GLkYwRixGMEYtRi9GLUYxRjBGM0YwRjU=

t1 est diff\351rent de t2 (on cherche un point double) et tous deux sont distincts de 1 (voir domaine de d\351finition)

Donc ce syst\350me est \351quivalent au nouveau syst\350me

systeme_equiv:=(t2-1)*(t1-1)/(t1-t2)*systeme;

NiM+SS5zeXN0ZW1lX2VxdWl2RzYiKiosJkkjdDJHRiUiIiIhIiJGKUYpLCZJI3QxR0YlRilGKkYpRiksJkYoRipGLEYpRio3JC8sJCouLCYqJkYoRilGLEYpRilGKUYpRilGLUYpRidGKiwmRihGKUYpRilGKkYrRiosJkYsRilGKUYpRipGKiIiIS8qKiwoRjNGKUYsRipGKEYqRilGLUYpRidGKkYrRipGNkYp

On met un peu d'ordre

systeme_equiv:=simplify(expand(systeme_equiv));

NiM+SS5zeXN0ZW1lX2VxdWl2RzYiNyQvLCQqKCwmKiZJI3QyR0YlIiIiSSN0MUdGJUYtRi1GLUYtRi0sJkYuRi1GLUYtISIiLCZGLEYtRi1GLUYwRjAiIiEvLChGK0YtRi5GMEYsRjBGMg==

Multiplier la premiere composante par (t1+1)(t2+1) nous donne un deuxi\350me systeme \351quivalent car t1 et t2 sont distincts de -1

Mult:=(a,b)->[(t1+1)*(t2+1)*a,b];

NiM+SSVNdWx0RzYiZio2JEkiYUdGJUkiYkdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQqKCwmSSN0MUdGJSIiIkYxRjFGMSwmSSN0MkdGJUYxRjFGMUYxOSRGMTklRiVGJUYl

systeme_bis:=Mult(op(systeme_equiv));

NiM+SSxzeXN0ZW1lX2Jpc0c2IjckLywmKiZJI3QyR0YlIiIiSSN0MUdGJUYrISIiRi1GKyIiIS8sKEYpRitGLEYtRipGLUYu

Au lieu de r\351soudre ce syst\350me en t1 et t2 nous allons le r\351soudre pour les inconnues p=t1*t2 et s=t1+t2

Sps:=subs(t2*t1=p,t1=s-t2,systeme_bis);

NiM+SSRTcHNHNiI8JC8sJkkicEdGJSIiIkkic0dGJSEiIiIiIS8sJkYpRixGLEYqRi0=

On change la structure de liste en structure d'ensemble

Sps:={op(Sps)};

NiM+SSRTcHNHNiI8JC8sJkkicEdGJSIiIkkic0dGJSEiIiIiIS8sJkYpRixGLEYqRi0=

solve(Sps);

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Et donc connaissant le produit p et la somme s, on sait que t1 et t2 sont les deux racines distinctes de X^2-sX+p=X^2+X-1... CQFD

4eme M\351thode

#Derni\350re M\351thode (promis c'est vraiment la derni\350re): On laisse MAPLE tout faire

systeme:={x(t1)-x(t2)=0,y(t1)-y(t2)=0};

NiM+SShzeXN0ZW1lRzYiPCQvLCYqJkkjdDFHRiUiIiIsJiokRioiIiNGKyEiIkYrRi9GKyomSSN0MkdGJUYrLCYqJEYxRi5GK0YvRitGL0YvIiIhLywmKiZGKkYuLCZGKkYrRi9GK0YvRisqJkYxRi4sJkYxRitGL0YrRi9GL0Y0

reponse:=[solve(systeme)];

NiM+SShyZXBvbnNlRzYiNyQ8JC9JI3QyR0YlRikvSSN0MUdGJUYpPCQvRiktSSdSb290T2ZHNiRJKnByb3RlY3RlZEdGMUkoX3N5c2xpYkdGJTYjLCgqJEkjX1pHRjAiIiMiIiJGNkY4ISIiRjgvRissJkY5RjhGLkY5

On exclut bien sur les solutions correspondants \340 des param\350tres identiques (on veut des points DOUBLES);

allvalues(reponse);

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