Parábola como envolvente de familia de tangentes que descansan equidistantes sobre los ejes.

Familia de tangentes

> restart;

> y=-(1+u)/(5-u)*x+(1+u);

> for i from 0 to 10 do

> u:=-0.9+0.58*i;

> p.i:=plot(-(1+u)/(5-u)*x+(1+u),x=0..5,color=blue,thickness=2):

> od:

> display([seq(p.i,i=0..10)],view=[0..5,0..5],scaling=constrained);

Derivada de familia respecto del parámetro y simplificaciones

> diff(-y-(1+u)/(5-u)*x+(1+u),u);

> simplify((5-u)^2*(-1/(5-u)*x-(1+u)/(5-u)^2*x+1));

> -(1+u)/(5-u)*x+(1+u);

> simplify((5-u)*(-y-(1+u)/(5-u)*x+1+u));

Obtención de envolvente por Bases de Groebner.

> restart;

> with(Groebner):

> WL:=[-6*x+25-10*u+u^2,-5*y+y*u-x-x*u+5+4*u-u^2];

> gbasis(WL,plex(u,y,x));

>

> simplify(1/x*(-12*x^2+36*x-12*y*x+y^2*x-2*y*x^2+x^3));

Verificación de parábola por completesquare

> restart;

> with(student):

> completesquare(-12*x+36-12*y+y^2-2*y*x+x^2,x);

X= ------->>> X^2 - 24*y -->>> parábola rotada

Plots de familia y curva envolvente

> restart;

> with(plots):

> for i from 0 to 10 do

> u:=-0.9+0.58*i;

> p.i:=plot(-(1+u)/(5-u)*x+(1+u),x=0..5,color=gray,thickness=2):

> od:

> display([seq(p.i,i=0..10),implicitplot(-12*x+36-12*y+y^2-2*y*x+x^2,x=0..5,y=0..5,color=blue,thickness=3)],view=[0..5,0..5],scaling=constrained);