GeometricLocusParabolicCylinder::usage="A point moves in such a way that its distance to the Y axis is always equal to its distance to the plane x - z = 1. Find and identify the equation of the geometric locus. References: Lehmann, Charles H. Geometría Analítica. Ed. Limusa: p.405, 2003; Personal.";
Applying the information:
In[]:=
Norm[{-x,-y,-z}{0,1,0}]
Norm[{0,1,0}]
Out[]=
2
Abs[x]
+
2
Abs[z]
x-z-1
2
=
2
x
+
2
z
In[]:=
Expand
2
(x-z-1)
-
2

2
2
x
+
2
z


1-2x-
2
x
+2z-2xz-
2
z
=0;
ℒ
1
=x1,x1,-x1+1+
2-4x1
+t{1,0,-1};​​
ℒ
2
=x1,x1,-x1+1-
2-4x1
+t{1,0,-1};
In[]:=
Solve[x1+t--x1+1+
2-4x1
-t==1,t]
t
1
2
2+
2
1-2x1
-2x1;
In[]:=
Expandx1+
1
2
2+
2-4x1
-2x1
Out[]=
1+
1
2
2-4x1
In[]:=
Expand-x1+1+
2-4x1
-
1
2
2+
2-4x1
-2x1
Out[]=
1
2
2-4x1
P1=1+
1
2
2-4x1
,x1,
1
2
2-4x1
;
In[]:=
Solve[x1+t--x1+1-
2-4x1
-t==1,t]
t
1
2
2-
2
1-2x1
-2x1;
In[]:=
Expandx1+
1
2
2-
2-4x1
-2x1
Out[]=
1-
1
2
2-4x1
In[]:=
Expand-x1+1-
2-4x1
-
1
2
2-
2-4x1
-2x1
Out[]=
-
1
2
2-4x1
P2=1-
1
2
2-4x1
,x1,-
1
2
2-4x1
;
In[]:=
Manipulated1=NAbs
2x1-
2-4x1
-2
2
;d2=NAbs
2x1+
2-4x1
-2
2
;d3=N
2
x1-1-
1
2
2-4x1
+
2
-x1+1+
1
2
2-4x1
;d4=N
2
x1-1+
1
2
2-4x1
+
2
-x1+1-
1
2
2-4x1
;​​w=Grid[
Pleaf1π
,
Pleaf1Y
,{d1,d3},FrameAll,ItemStyle->15,ItemSize->15];​​q=Grid[
Pleaf2π
,
Pleaf2Y
,{d2,d4},FrameAll,ItemStyle->15,ItemSize15];​​ShowContourPlot3D[1-2x-
2
x
+2z-2xz-
2
z
==0,{x,-10,10},{y,-10,10},{z,-10,10},Mesh->None,ContourStyleDirective[Blue,Opacity[0.3],Specularity[White,30]],PerformanceGoal"Quality",PlotPoints->60],ContourPlot3D[x-z==1,{x,-5,5},{y,-5,5},{z,-5,5},Mesh->None,ContourStyleDirective[Purple,Opacity[0.3],Specularity[White,30]],PerformanceGoal"Quality"],Graphics3D[Red,Ball[x1,x1,-x1+1+
2-4x1
,0.1]],Graphics3D[Red,Ball[x1,x1,-x1+1-
2-4x1
,0.1]],Graphics3DBlack,Ball1+
1
2
2-4x1
,x1,
1
2
2-4x1
,0.1,Graphics3DBlack,Ball1-
1
2
2-4x1
,x1,-
1
2
2-4x1
,0.1,​​Graphics3D[{Black,Ball[{0,x1,0},0.1]}],Graphics3DBlack,Dashed,Thick,Linex1,x1,-x1+1+
2-4x1
,1+
1
2
2-4x1
,x1,
1
2
2-4x1
,Graphics3DBlack,Dashed,Thick,Linex1,x1,-x1+1-
2-4x1
,1-
1
2
2-4x1
,x1,-
1
2
2-4x1
,Graphics3D[Black,Dashed,Thick,Line[x1,x1,-x1+1+
2-4x1
,{0,x1,0}]],Graphics3D[Black,Dashed,Thick,Line[x1,x1,-x1+1-
2-4x1
,{0,x1,0}]],AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSizeFull,ViewPoint{-2,-2,0},Style["Parabolic cylinder:",Bold,Large],{x1,0,"Value (​
x
1
​)"},-5,
1
2
,0.01,Delimiter,{{w,2,"Distance of the point
P
leaf1
to the plane π and the point
P
leaf1
to the Y axis"}},Delimiter,{{q,4,"Distance of the point
P
leaf2
to the plane π and the point
P
leaf2
to the Y axis"}},ControlPlacementTop
Out[]=
​
Parabolic cylinder:
Value (​
x
1
​)
Distance of the point
P
leaf1
to the plane π and the point
P
leaf1
to the Y axis
Pleaf1π
Pleaf1Y
9.47468
9.47468
Distance of the point
P
leaf2
to the plane π and the point
P
leaf2
to the Y axis
Pleaf2π
Pleaf2Y
3.70579
3.70579