On joint un point M d'une ellipse aux foyers F1 et F2. Les droites MF1 et MF2 recoupent l'ellipse aux points H1 et H2 , trouver l'enveloppe de la droite H1H2, quand le point M se `déplace` sur l'ellipse.;
restart;
Fig := proc(t) local a, b, c, courbe, sol, sol1, dr, tx; _EnvHorizontalName := 'x'; _EnvVerticalName := 'y'; a := 11; b := 7; c := sqrt(a^2 - b^2); geometry:-ellipse(e1, x^2/a^2 + y^2/b^2 = 1); geometry:-point(Oo, 0, 0); geometry:-point(M, a*cos(t), b*sin(t)); geometry:-point(F1, -c, 0); geometry:-point(F2, c, 0); geometry:-line(MF1, [M, F1]); geometry:-line(MF2, [M, F2]); sol := solve({geometry:-Equation(MF1), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H1, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH1, [M, H1]); sol := solve({geometry:-Equation(MF2), x^2/a^2 + y^2/b^2 = 1}, {x, y}, explicit); print(%); geometry:-point(H2, subs(sol[2], x), subs(sol[2], y)); geometry:-line(MH2, [M, H2]); courbe := plots:-implicitplot(x^2/a^2 + (a^2 + c^2)^2*y^2/b^2 - 1 = 0, x = -a .. a, y = -b .. b, color = cyan); tx := plots:-textplot([[geometry:-coordinates(M)[], "M"], [geometry:-coordinates(Oo)[], "O"], [geometry:-coordinates(H1)[], "H1"], [geometry:-coordinates(H2)[], "H2"], [geometry:-coordinates(F1)[], "F1"], [geometry:-coordinates(F2)[], "F2"]], font = [times, bold, 16], align = [above, left]); dr := geometry:-draw([e1(color = blue), MH1(color = magenta), MH2(color = magenta), M(color = red, symbol = solidcircle, symbolsize = 12), H1(color = red, symbol = solidcircle, symbolsize = 12), H2(color = red, symbol = solidcircle, symbolsize = 12), F1(color = red, symbol = solidcircle, symbolsize = 12), F2(color = red, symbol = solidcircle, symbolsize = 12), Oo(color = red, symbol = solidcircle, symbolsize = 12)]); plots:-display([dr, tx, courbe], scaling = constrained, axes = normal, title = "Ellipse et normales ", titlefont = [HELVETICA, 14]); end proc;
Fig(Pi/3);
/ 11 7 (1/2)\ / 26411 210177 (1/2)
{ x = --, y = - 3 }, { x = - ----- - ------ 2 ,
\ 2 2 / \ 57074 28537
11319 (1/2) (1/2) 66199 (1/2)\
y = ----- 3 2 - ----- 3 }
28537 57074 /
/ 11 7 (1/2)\ / 210177 (1/2) 26411
{ x = --, y = - 3 }, { x = ------ 2 - -----,
\ 2 2 / \ 28537 57074
11319 (1/2) (1/2) 66199 (1/2)\
y = - ----- 3 2 - ----- 3 }
28537 57074 /
Fig(Pi/6);
/ 104027 (1/2) 17787 (1/2) (1/2) 123420 (1/2)
{ x = - ------ 3 + ----- 3 6 - ------ 2
\ 22226 11113 11113
19404 (1/2) (1/2) 66199 11319 (1/2)\
+ ----- 2 6 , y = - ----- + ----- 6 },
11113 22226 11113 /
/ 11 (1/2) 7\
{ x = -- 3 , y = - }
\ 2 2/
Error, (in geometry:-line) the line is not defined
plots:-animate(Fig, [t], t = 0.1 .. 2*Pi, frames = 150);
{x = -10.99908244, y = -0.09041172732},
{x = 10.94504582, y = 0.6988339166}
Error, (in plots/animate) the line is not defined
;
NULL;
Thank you for your help.