intersections := proc(P, Q, T)
local R, W, w, t, a, b, sol, buff, v;
sol := []; if T = Y then W := X; else W := Y; end if;
R := resultant(P, Q, T);
print(`Résultant :`);
print(R);
w := fsolve(R, W); t := [];
for v in [w] do t := [op(t), fsolve(subs(W = v, P), T)]; end do;
for a in [w] do for b in [t] do if T = Y then buff := abs(subs(X = a, Y = b, P)) + abs(subs(X = a, Y = b, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, a, b, buff); if buff < 1/100000000 then sol := [op(sol), [a, b]]; end if;
else buff := abs(subs(X = b, Y = a, P)) + abs(subs(X = b, Y = a, Q));
printf(`X=%a,   Y=%a   --->  %a\\n`, b, a, buff); if buff < 1/100000000 then sol := [op(sol), [b, a]];
end if; end if; end do; end do;
printf(`Nombre de solutions :  %a\\n`, nops(sol)); print(sol); end proc;
Try with :
intersections(X^2 + Y^2 - 1, X - Y, X); Would you like to develop this procedure which does not give the number of solutions ?Thank you.

On considère une ellipse x^2/a^2+y^2/b^2-1=0 et 2 sommets de cette ellipse A(a,0) et B(0,b). On imagine une hyperbole équilatère variable passant par les points O, A et B. Cette courbe rencontre l'ellipse en 2 autres points A1 et B1. Montrer que la droite A1B1 passe par un point fixe. Même avec l'intelligence artificielle, je ne parviens pas à résoudre ce problème. Pourriez-vous d'aider. Merci.

Machine translation by moderator:

We consider an ellipse x^2/a^2+y^2/b^2-1=0 and 2 vertices of this ellipse A(a,0) and B(0,b). We imagine a variable equilateral hyperbola passing through the points O, A and B. This curve meets the ellipse at 2 other points A1 and B1. Show that the line A1B1 passes through a fixed point. Even with artificial intelligence, I can't solve this problem. Could you help. Thank you.

On donne une ellipse rapportée à ses axes x^2/a^2+y^2/b^2-1=0 et une droite (D) qui rencontre cette
courbe en 2 points A et B. 
On considère un cercle variable passant parles points A et B et on demande le lieu géométrique des points de rencontre des tangentes communes au cercle et à l'ellipse.
restart;
with(plots);
with(VectorCalculus);
a := 5;
b := 3;
ellipse_eq := (x, y) -> x^2/a^2 + y^2/b^2 - 1;
m := 1;
c := -2;
line_eq := (x, y) -> y - m*x - c;
intersections := solve({line_eq(x, y) = 0, ellipse_eq(x, y) = 0}, {x, y}, explicit);
A := intersections[1];
B := intersections[2];
A := [VectorCalculus:-`+`(VectorCalculus:-`*`(25, 17^VectorCalculus:-`-`(1)), VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1))), VectorCalculus:-`+`(VectorCalculus:-`-`(VectorCalculus:-`*`(9, 17^VectorCalculus:-`-`(1))), VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1)))];
B := [VectorCalculus:-`+`(VectorCalculus:-`*`(25, 17^VectorCalculus:-`-`(1)), VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1)))), VectorCalculus:-`+`(VectorCalculus:-`-`(VectorCalculus:-`*`(9, 17^VectorCalculus:-`-`(1))), VectorCalculus:-`-`(VectorCalculus:-`*`(VectorCalculus:-`*`(15, sqrt(30)), 34^VectorCalculus:-`-`(1))))];
center_x := VectorCalculus:-`*`(VectorCalculus:-`+`(A[1], B[1]), 2^VectorCalculus:-`-`(1));
center_y := VectorCalculus:-`*`(VectorCalculus:-`+`(A[2], B[2]), 2^VectorCalculus:-`-`(1));
radius := VectorCalculus:-`*`(sqrt(VectorCalculus:-`+`(VectorCalculus:-`+`(A[1], VectorCalculus:-`-`(B[1]))^2, VectorCalculus:-`+`(A[2], VectorCalculus:-`-`(B[2]))^2)), 2^VectorCalculus:-`-`(1));
circle_eq := (x, y) -> (x - center_x)^2 + (y - center_y)^2 - radius^2;
L := (x1, y1, x2, y2, lambda1, lambda2) -> (x1 - x2)^2 + (y1 - y2)^2 + lambda1*ellipse_eq(x1, y1) + lambda2*circle_eq(x2, y2);
eq1 := diff(L(x1, y1, x2, y2, lambda1, lambda2), x1);
eq2 := diff(L(x1, y1, x2, y2, lambda1, lambda2), y1);
eq3 := diff(L(x1, y1, x2, y2, lambda1, lambda2), x2);
eq4 := diff(L(x1, y1, x2, y2, lambda1, lambda2), y2);
eq5 := ellipse_eq(x1, y1);
eq6 := circle_eq(x2, y2);
sols := solve({eq1, eq2, eq3, eq4, eq5, eq6}, {lambda1, lambda2, x1, x2, y1, y2}, explicit);
sols;
lieu_geometrique := [seq([sols[i][1], sols[i][2]], i = 1 .. nops(sols))];
plot(lieu_geometrique, style = point, symbol = cross, color = red, title = "Lieu géométrique des points de rencontre");
Ce code m'a été donné en partie par l'intelligence artificielle (Mistral), mais il se plante. Pourriez-vous corriger les erreurs. Merci.

On considère un cercle fixe O et un point fixe A extérieur. Une sécante variable BC à ce cercle passe par un point fixe J.
Démontrer que le cercle ABC passe par un second point fixe P.
restart;
Proc := proc(m)
local xA, yA, xB, yB, xC, yC, xJ, yJ, tx, dr, Oo, c1, r, eqBJ, eq1, sol;
_EnvHorizontalName := 'x'; _EnvVerticalName := 'y';
xJ := 5; yJ := 1; geometry:-point(A, 2, 4); geometry:-point(J, xJ, yJ); geometry:-point(Oo, 0, 0);
r := 3; c1 := plottools[geometry:-circle]([0, 0], r, color = blue);
eqBJ := y = m*(x - xJ) + yJ; geometry:-line(BJ, eqBJ, [x, y]);
eq1 := x^2 + y^2 = r^2; sol := solve({eqBJ, eq1}, {x, y}, explicit);
xB := subs(sol[1], x); yB := subs(sol[1], y);
geometry:-point(B, xB, yB); xC := subs(sol[2], x); yC := subs(sol[2], y);
geometry:-point(C, xC, yC); geometry:-circle(c2, [A, B, C]); geometry:-line(AB, [A, B]); geometry:-line(AC, [A, C]);
eqBJ := y = m*(x - xJ) + yJ; geometry:-line(BJ, eqBJ, [x, y]);
eq1 := x^2 + y^2 = r^2; sol := solve({eqBJ, eq1}, {x, y},explicit);
xB := subs(sol[1], x); yB := subs(sol[1], y); geometry:-point(B, xB, yB);
xC := subs(sol[2], x); yC := subs(sol[2], y); geometry:-point(C, xC, yC);
geometry:-circle(c2, [A, B, C]); geometry:-line(AB, [A, B]); geometry:-line(AC, [A, C]);
tx := plots:-textplot([[geometry:-coordinates(A)[], "A"], [geometry:-coordinates(B)[], "B"], [geometry:-coordinates(C)[], "C"], [geometry:-coordinates(J)[], "J"]], font = [times, bold, 16], align = [above, right]);
dr := geometry:-draw([AB(color = black), c2(color = magenta), A(color = blue, symbol = solidcircle, symbolsize = 16),
B(color = red, symbol = solidcircle, symbolsize = 16), C(color = red, symbol = solidcircle, symbolsize = 16),
J(color = red, symbol = solidcircle, symbolsize = 16)]); plots:-display([dr, c1, tx], axes = normal, view = [-5 .. 6, -4 .. 6], scaling = constrained);
end proc;
plots:-animate(Proc, [m], m = -0.9 .. 0.2*Pi, frames = 50);
Error, (in plots/animate) two lists or Vectors of numerical values expected
NULL;
I am trying to find out point P; Thank you for your help.

OneFrame := proc(k)
local Courbe, T, a, b, c, t, P, Q, NormM, F, Ell, sol, N1, N2, dr, tx;
a := 11; b := 7; c := sqrt(a^2 - b^2); t := 1/3*Pi;
Ell := x^2/a^2 + y^2/b^2 = 1;
geometry:-point(T, (a^2 - b^2)*cos(t)^3/a, -(a^2 - b^2)*sin(t)^3/b);
Courbe := plots:-implicitplot(Ell, x = -a - 10 .. a + 10, y = -b - 10 .. b + 10, scaling = constrained, color = blue);
NormM := plots:-implicitplot(y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), x = -a - 5 .. a + 10, y = -b - 10 .. b + 10, color = orange); geometry:-line(Per, y - b*sin(t) = a*sin(t)*(x - a*cos(t))/(b*cos(t)), [x, y]);
geometry:-point(P, subs(y = 0
, geometry:-Equation(Per), 0));
geometry:-point(Q, 0, subs(x = 0, geometry:-Equation(Per)));
geometry:-point(M, a*cos(t), b*sin(t));
geometry:-point(N1, a*cos(k), b*sin(k));
geometry:-point(F, 2.329411765, -2.567510609);
geometry:-line(L, N1, F);
sol := solve({geometry:-Equation(L), Ell}, {x, y},explicit);
geometry:-point(N2, subs(sol[2], x), subs(sol[2], y));
geometry:-segment(sg, N1, N2);
tx := plots:-textplot([[geometry:-coordinates(M)[], "M"],
[geometry:-coordinates(N1)[], "N1"], [geometry:-coordinates(N2)[], "N2"],
[geometry:-coordinates(P)[], "P"],
[geometry:-coordinates(Q)[], "Q"],
[geometry:-coordinates(F)[], "F point de Frégier"],
[geometry:-coordinates(T)[], "T"]], font = [times, bold, 16], align = [above, left]);
dr := geometry:-draw([sg(color = magenta, linestyle = dash),
Per(color = black), P(color = red, symbol = solidcircle, symbolsize = 12),
Q(color = red, symbol = solidcircle, symbolsize = 12),
M(color = black, symbol = solidcircle, symbolsize = 12),
F(color = red, symbol = solidcircle, symbolsize = 12),
N1(color = black, symbol = solidcircle, symbolsize = 8),
N2(color = black, symbol = solidcircle, symbolsize = 8),
T(color = black, symbol = solidcircle, symbolsize = 8)]);
plots:-display(Courbe, tx, dr, scaling = constrained, axes = none); end proc;

plots:-animate(OneFrame, [k], k = Pi/3 .. Pi, frames = 50);
Error, (in plots/animate) wrong type of arguments
Why this animation does't work ? Thank you very much.