Exercice 3On d\351fnit la param\351trisation, puis on construit le support partiel et sir la totalit\351 des param\350tresrestart;x:=t->3*t^2;y:=t->2*t^3;

NiM+SSJ4RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJDkkIiIjIiIkRiVGJUYlNiM+SSJ5RzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCQqJDkkIiIkIiIjRiVGJUYl

total:=[x(t),y(t),t=-100..100]: plot(total);

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Calculons le vecteur vitesse VV:=unapply([D(x)(t),D(y)(t)],t);

NiM+SSJWRzYiZio2I0kidEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlNyQsJDkkIiInLCQqJEYuIiIjRi9GJUYlRiU=

M\352me pour M(0) la tangente en M(t) est Y=t(X-x(t))+y(t)Cherchons les points d'intersection de Tt avec Gammaequ:=y(tau)=t*(x(tau)-x(t))+y(t);solve(equ,tau);

NiM+SSRlcXVHNiIvLCQqJEkidEdGJSIiJCMhIiIiIiVGJw==NiMsJEkidEc2IiMhIiIiIiM=

tau:=-t/2;

NiM+SSR0YXVHNiIsJEkidEdGJSMhIiIiIiM=

On veut que la tangente en M(tau) soit orthogonale \340 la tangente en M(t), i.e vu que les pentes sont tau et teq:=tau*t=-1;solve(eq,t);t:='t':

NiM+SSNlcUc2Ii8sJCokSSJ0R0YlIiIjIyEiIkYqRiw=NiQsJCokIiIjIyIiIkYlISIiRiQ=

2 solutions:logique car on a une sym\351trie / OxOn \351crit la tangente en t0=2^1/2 et la tangente en tau_0=-2^1/2/2 (et le sym\351trique)t_0:=sqrt(2):xT1:=x(t_0)+t:yT1:=y(t_0)+t_0*t:EqT1:=[xT1,yT1,t=-infinity..infinity]:tau_0:=-sqrt(2)/2:xN1:=x(tau_0)+t:yN1:=y(tau_0)+tau_0*t:EqN1:=[xN1,yN1,t=-infinity..infinity]:t_0:=-sqrt(2):xT1:=x(t_0)+t:yT1:=y(t_0)+t_0*t:EqT2:=[xT2,yT2,t=-infinity..infinity]:tau_0:=sqrt(2)/2:xN2:=t:yN2:=tau_0*(t-x(tau_0))+y(tau_0):EqN2:=[xN2,yN2,t=-infinity..infinity]:total:=[x(t),y(t),t=-infinity..infinity]: plot([total,EqT1,EqN1,EqT2,EqN2],color=[red,blue,blue,green,green],thickness=[2,2,1,2,1]);

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