270 Reputation

4 years, 166 days

How to show 2 expressions are equivalent...

Maple 2018

A line is deternined on one side by 16*x-2*y-11*z=0, 14*x-y-10*z=3 and on the other by (x-2)/3=(y-5)/2=(z-2)/4. Are they equivalent  ?  Thank you.

how to draw these 3 lines and then proje...

Maple 2018

how to draw these 3 lines and then project them on the plans Oxy,Oxz,Oyz;
3 given lines a := [3*t-7, -2*t+4, 3*t+4]; b := [m+1, 2*m-9, -m-12];c:={x = -200/29-2*t, y = 114/29+3*t, z = 119/29+4*t}, how to show these lines and the projections on the 3 planes ? Thank you.

How to show that these 2 procedures are ...

Maple 2018

A := [1, -2, 3]:u := `<,>`(0, -2, 2):v := `<,>`(5, 8, -3):
PL := proc (A, u, v) local d, Det, AP, t, U, V;
AP := `<,>`(x-A[1], y-A[2], z-A[3]);
Matrix(`<|>`(`<,>`(AP), `<,>`(u), `<,>`(v)));
Det := LinearAlgebra:-Determinant(%); d := igcd(coeff(Det, x), coeff(Det, y), coeff(Det, z), tcoeff(Det));
print(`Une équation cartésienne du plan est :`);
t := Det/d; print([t = 0]);
print('Une*représentation*paramétrique*du*plan*est; -1');
U := convert(u, list); V := convert(v, list);
[x = lambda*U[1]+mu*V[1]+A[1], y = lambda*U[2]+mu*V[2]+A[2], z = lambda*U[3]+mu*V[3]+A[3]] end proc;
PL(A, u, v);

plan3p := proc (A::list, B::list, C::list)
local d, M, N, P, Mat, Det, t, U, V;
M := `<,>`(x-A[1], B[1]-C[1], C[1]-A[1]); N := `<,>`(y-A[2], B[2]-C[2], C[2]-A[2]); P := `<,>`(z-A[3], B[3]-C[3], C[3]-A[3]);
Mat := Matrix([M, N, P]);
Det := LinearAlgebra:-Determinant(%);
d := igcd(coeff(Det, x), coeff(Det, y), coeff(Det, z), tcoeff(Det));
print(`Une équation cartésienne du plan est :`);
t := Det/d; print([t = 0]); print('Une*représentation*paramétrique*du*plan*est; -1');
U := A-B; V := B-C;
[x = lambda*U[1]+mu*V[1]+A[1], y = lambda*U[2]+mu*V[2]+A[2], z = lambda*U[3]+mu*V[3]+A[3]] end proc;
A := [-6, 3, -2]; B := [5, 2, 1]; C := [2, 5, 2];plan3p(A, B, C);
How to know if these procedures are correct or not. Thank you.

How to find an orthogonal circle ? ...

Maple 2018

We consider a fixed circle (C) tangent to a fixed line Δ at a given point O of this line.
Circles Γ tangent to circles C in M and to the right Δ in N are studied.
Show that the MN line passes through a fixed point I. Infer that the circles Γ remain orthogonal to a fixed circle.
My code is :
restart; with(geometry);with(plots);
_EnvHorizontalName := 'x';_EnvVerticalName := 'y';
dist := proc (M, N) sqrt(Vdot(expand(M-N), expand(M-N))) end proc;
point(oo, 0, 3); p := 6;
point(N, 5, 0);
line(Delta, y = 0, [x, y]);
para := x^2 = 2*p*y;
solve(subs(x = 5, para), y); point(varpi, 5, 25/12);
line(alpha, [oo, varpi]); k := 3/(25/12);
point(M, (0+5*k)/(1+k), (3+25*k*(1/12))/(1+k));
circle(C, x^2+(y-3)^2 = 9, [x, y]);cir := implicitplot(x^2+(y-3)^2 = 9, x = -5 .. 5, y = -5 .. 7, color = blue);
Para := implicitplot(para, x = -40 .. 40, y = 0 .. 40, linestyle = 3, color = coral);
homothety(J, N, -k, M); coordinates(J);
circle(C1, (y-25/12)^2+(x-5)^2 = (25/12)^2, [x, y]);line(lNJ, [N, J]);
triangle(T1, [J, oo, M]); triangle(T2, [N, varpi, M]);
C1 := implicitplot((y-25/12)^2+(x-5)^2 = (25/12)^2, x = 2 .. 8, y = 0 .. 5, color = magenta);dr1 := draw([oo, Delta, varpi, N, M, J], printtext = true); dr2 := draw([alpha(color = black), lNJ(color = black), T1(color = green, filled = true), T2(color = green, filled = true)]);
inversion(M, M, C);
inversion(N, M, C);
Fig := proc (xOm)
local cir, c2, C2, C1, c3, C3, k, M, N, J, sol, dr, varpi;
global p, para, Para;
sol := solve(subs(x = xOm, para), y);
cir := (y-sol)^2+(x-xOm)^2 = sol^2; c2 := x^2+(y-3)^2 = 9;
geometry:-point(N, xOm, 0); sol := solve(subs(x = xOm, para), y);
geometry:-point(varpi, xOm, sol); k := 3/sol;
geometry:-point(M, xOm*k/(k+1), (3+k*sol)/(k+1));
geometry:-homothety(J, N, -k, M);
c3 := (x-(1/2)*xOm)^2+(y-3)^2 = (1/4)*dist(N, J)^2;
C1 := plots:-implicitplot(cir, x = -xOm .. 3*xOm, y = 0 .. 3*xOm, color = magenta);
C2 := plots:-implicitplot(c2, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);
C3 := plots:-implicitplot(c3, x = -xOm .. 2*xOm, y = 0 .. 2*xOm, color = blue);
dr := geometry:-draw([varpi, M, J]);
plots:-display([Para, C2, C1, C3, dr], view = [-xOm .. 3*xOm, -1 .. 3*xOm], axes = normal, scaling = constrained) end proc;

Fig(8);
display([seq(Fig(4+.8*i), i = 4 .. 15)]);
display({C1, Para, cir, dr1, dr2}, view = [-8 .. 8, -1 .. 8], axes = normal, scaling = constrained, size = [500, 500]);
I don't know what is that orthogonal circle to each tangent circles. Thank you to help me.

.

Why Explore does't work ?...

Maple 2018

restart;
with(plots); with(LinearAlgebra);
_EnvHorizontalName := 'x';

_EnvVerticalName := 'y';

x1,y1,x2,y2,x3,y3:=0,-3,3,1,5,-2:
A := [x1, y1]: B := [x2, y2]: C := [x3, y3]:

Barycentre := proc (A, B, t) description "Barycentre de 2 points A(1) et B(t) dans le rapport t";
return [(1-t)*A[1]+t*B[1], (1-t)*A[2]+t*B[2]] end proc;
ellip := proc (r1, r2) local a, b, c, d, e, f, D, E, F, eq1, eq2, eq3, eq4, eq5, eq6, x0, y0, EE, r3, sol, Ff, Tg;
global A, B, C;
r3 := -1/(r2*r1);
D := Barycentre(C, B, 1/(1-r1)); E := Barycentre(A, C, 1/(1-r2)); F := Barycentre(B, A, 1/(1-r3));
Ff := proc (x, y) options operator, arrow; a*x^2+2*b*x*y+c*y^2+2*d*x+2*e*y+f end proc;
Tg := proc (x0, y0, x, y) options operator, arrow; a*x*x0+b*(x*y0+y*x0)+c*y*y0+d*(x+x0)+e*(y+y0)+f end proc;
eq1 := Ff(D[1], D[2]);
eq2 := Ff(E[1], E[2]);
eq3 := Ff(F[1], F[2]);
eq4 := Tg(F[1], F[2], x1, y1);
eq5 := Tg(D[1], D[2], x2, y2);
eq6 := Tg(E[1], E[2], x3, y3);
sol := op(solve([eq1, eq2, eq3, eq4, eq5, eq6], [a, b, c, d, e]));
assign(sol);
EE := subs(f = 1, Ff(x, y) = 0) end proc;

ellip(-1, -7); tri := plot([A, B, C, A], color = blue):

po := plot([A, B, C], style = point, symbolsize = 15, symbol = solidcircle, color = red);
tp := textplot([[A[], "A"], [B[], "B"], [C[], "C"]], 'align' = {'above', 'left'});
x := 'x'; y := 'y';
ELL := seq(implicitplot(ellip(-7/11-(1/11)*j, -1/17-3*j*(1/17)), x = 0 .. 5, y = -3 .. 1, color = ColorTools:-Color([rand()/10^12, rand()/10^12, rand()/10^12])), j = 1 .. 17);
display([tri, ELL, po, tp], view = [-.5 .. 5.5, -4 .. 1.5], axes = none, scaling = constrained, size = [500, 500]);
Explore(implicitplot(ellip(r1, r2), x = 0 .. 5, y = -3 .. 1), parameters = [r1 = -2.18 .. -.7, r2 = -3 .. -.23]);
Can you tell me why this last instruction does't work ? Thank you.

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