> 
O14-903
> restart;with(plots):
> r:=t->p/(1+e*cos(t));

                                       p
                        r := t -> ------------
                                  1 + e cos(t)

> trac:=proc(a,b)
> plot([subs({p=a,e=b},r(t)),t,t=-Pi..Pi],-5..5,-4..4,coords=polar,scaling=constrained,color=black) 
> end;

  trac := proc(a, b)
    plot([subs({p = a, e = b}, r(t)), t, t = -Pi .. Pi],
    -5 .. 5, -4 .. 4, coords = polar, scaling = constrained,
    color = black)
end

> trac(1,.8);


> tanV:=r/D(r);

                                      r
                     tanV := --------------------
                                    p e sin(t)
                             t -> ---------------
                                                2
                                  (1 + e cos(t))

> # La tangente au point d'angle polaire u est y=y0+tan(u+V)(x-x0) oł tan(V)=r/r'
> tang:=proc(a,b,u)
> local dte,x0,y0;
> x0:=subs({p=a,e=b},r(u))*cos(u);
> y0:=subs({p=a,e=b},r(u))*sin(u);
> dte:=y0+(tan(u)+subs({p=a,e=b},tanV(u)))*(x-x0)/(1-tan(u)*subs({p=a,e=b},tanV(u)));
> end:
> display([trac(1,.8),plot(tang(1,.8,Pi/4),x=-5..5),plot(tang(1,.8,5*Pi/4),x=-5..5)]);


> sys:={y=tang(p,e,t),y=tang(p,e,t+Pi)};# Les tangentes en M1 et M2

  sys := {y =

                         /         1 - e cos(t)\ /      p cos(t)  \
                         |tan(t) - ------------| |x + ------------|
            p sin(t)     \           e sin(t)  / \    1 - e cos(t)/
        - ------------ + ------------------------------------------,
          1 - e cos(t)               tan(t) (1 - e cos(t))
                                 1 + ---------------------
                                           e sin(t)

                           /         1 + e cos(t)\ /      p cos(t)  \
                           |tan(t) + ------------| |x - ------------|
              p sin(t)     \           e sin(t)  / \    1 + e cos(t)/
        y = ------------ + ------------------------------------------
            1 + e cos(t)               tan(t) (1 + e cos(t))
                                   1 - ---------------------
                                             e sin(t)

        }

> solve(sys,{x,y});vert:=plot([1/(.8),s,s=-5..5],color=blue):# Les points N sur la droite (verticale) x=p/e

                                       2
                             p (-sin(t)  e + cos(t) + e)
             {x = p/e, y = - ---------------------------}
                               e sin(t) (1 + e cos(t))

> display([vert,trac(1,.8),seq(plot(tang(1,.8,k*Pi/20),x=-5..5),k=3..8),seq(plot(tang(1,.8,Pi+k*Pi/20),x=-5..5),k=3..8)]);

>