GeometricLocusMidpoints::usage="Find the geometric locus of the midpoints of the segments of length 5, one end of which describes the line y = 0, z = 0 and the other end, the line z = 3, y = 2x.References: Usunáriz Balanzategui, Ubaldo et al. Problemas de Geometría Analítica y Diferencial. p.230; Personal.";
Development:
Be the extreme points of the segment {a, 0 , 0} y {b, 2b, 3}:
In[]:=
M,b,;b=y;Solve==x,a
a+b
2
3
2
a+y
2
{{a2x-y}};
In[]:=
2
++
2
(a-b)
2
(0-2b)
2
(0-3)
2
(a-b)
2
b
In[]:=
Expand[-16++40]
2
(2x-2y)
2
y
Out[]=
-16+4-8xy+80
2
x
2
y
-4+-2xy+2==0;z==;
2
x
2
y
3
2
In[]:=
a1=1;b1=-2;c1=2;Indicador==-4a1c1
2
b1
Indicador-4;
It is an ellipse in space:
Its parametric equations are:
φ[x_]:=x,,;[x_]:=x,,;
2x+
32-4
2
x
4
3
2
φ
1
2x-
32-4
2
x
4
3
2
Graph :
In[]:=
ManipulateClearAll;φ[x_]:=x,,;[x_]:=x,,;[x_]:=2x-,0,0;[x_]:=,2,3;[x_]:=2x-,0,0;[x_]:=,2,3;p=Grid[{{{,,},{,,},LongitudeSegment1},{[t1],[t1],Norm[[t1]-[t1]]}},FrameAll,ItemSize->15,ItemStyle->15];q=Grid[{{{,,},{,,},LongitudeSegment2},{[t1],[t1],Norm[[t1]-[t1]]}},FrameAll,ItemSize->15,ItemStyle->15];ShowParametricPlot3D[φ[x],x,-2[x],x,-2[x],x,-2[x],x,-2[t1],0.09]}],Graphics3D[{Black,Ball[[t1],0.09]}],ParametricPlot3D[x],x,-2[x],x,-2[t1],0.09]}],Graphics3D[{Black,Ball[[t1],0.09]}],Graphics3D[{Black,Ball[[t1],0.09]}],Graphics3D[{Black,Thick,Dashed,Line[{[t1],[t1]}]}],Graphics3D[{Black,Thick,Dashed,Line[{[t1],[t1]}]}],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRange5,ViewPoint{-3,3,2},ImageSize900,Style["Geometric locus of the midpoint of segment:",Bold,Large],{t1,-2.828426,"Value (t1)"},-2
2x+
32-4
2
x
4
3
2
φ
1
2x-
32-4
2
x
4
3
2
ζ
1
2x+
32-4
2
x
4
ζ
2
2x+
32-4
2
x
4
2x+
32-4
2
x
4
ζ
3
2x-
32-4
2
x
4
ζ
4
2x-
32-4
2
x
4
2x-
32-4
2
x
4
x
1
y
1
z
1
x
2
y
2
z
2
ζ
1
ζ
2
ζ
1
ζ
2
x
3
y
3
z
3
x
4
y
4
z
4
ζ
3
ζ
4
ζ
3
ζ
4
2
,t1,PlotStyle->{Blue,Thick}],ParametricPlot3D[φ
1
2
,t1,PlotStyle->{Blue,Thick}],ParametricPlot3D[ζ
1
2
,t1,PlotStyle->{Black,Thick}],ParametricPlot3D[ζ
2
2
,t1,PlotStyle->{Black,Thick}],Graphics3D[{Red,Ball[φ[t1],0.09]}],Graphics3D[{Black,Ball[ζ
1
ζ
2
ζ
3
2
,t1,PlotStyle->{Black,Thick},ParametricPlot3D[ζ
4
2
,t1,PlotStyle->{Black,Thick}],Graphics3D[{Red,Ball[φ
1
ζ
3
ζ
4
ζ
1
ζ
2
ζ
3
ζ
4
2
+0.00000001,22
-0.00000001,0.01,Delimiter,{{p,2,"First segment data"}},Delimiter,{{q,2,"Second segment data"}},ControlPlacementTopOut[]=
