CylinderCircumscribedSphere::usage="Given the sphere f[x, y, z] =
2
x
+
2
y
+
2
z
- 4x - 6y - 8z = 0, to find 1) Tangent planes parallel to the plane 2x + 3y + 4z = 0. 2) Circumscribed cylinder of parallel generatrix to x = y = z. 3) Center and radius. References: Usunáriz Balanzategui, Ubaldo and Usunáriz Sala, Ignacio. Problemas de Geometría Analítica y Diferencial.​​ Prob.G63; Personal.";
Development:
1) Tangent planes parallel to the plane 2x + 3y + 4z = 0:
Method 1:
The sphere and the plane pass through the origin, the equation from the normal to the plane is:
ℒ:t{2,3,4}
Substituting in the equation of the sphere:
In[]:=
Solve[
2
(2t)
+
2
(3t)
+
2
(4t)
-4(2t)-6(3t)-8(4t)==0,t]
Out[]=
{{t0},{t2}}
The points are:
{0,0,0};​​{4,6,8};
One is the original plane and the other:
In[]:=
Solve[2(4)+3(6)+4(8)+d==0,d]
Out[]=
{{d-58}}
2x+3y+4z-58==0;
Method 2:
In[]:=
f[x_,y_,z_,t_]:=
2
x
+
2
y
+
2
z
-4xt-6yt-8zt;​​
f
x
[x_,y_,z_,t_]=D[f[x,y,z,t],x]​​
f
y
[x_,y_,z_,t_]=D[f[x,y,z,t],y]​​
f
z
[x_,y_,z_,t_]=D[f[x,y,z,t],z]​​
f
t
[x_,y_,z_,t_]=D[f[x,y,z,t],t]​​
Out[]=
-4t+2x
Out[]=
-6t+2y
Out[]=
-8t+2z
Out[]=
-4x-6y-8z
The normal line to the parallel planes is:
2x-4t
2
==
2y-6t
3
==
2z-8t
4
;
In[]:=
Reduce
2x-4t
2
==
2y-6t
3
==
2z-8t
4
,
2
x
+
2
y
+
2
z
-4xt-6yt-8zt==0,{x,y,z,t}
(x0&&y0&&z0)||y
3x
2
&&z2x&&x≠0&&t
x
4
;
Taking to x = 4 :
{0,0,0,1};​​{4,6,8,1};
To obtain the planes parallel to the original plan we consider 2x + 3y + 4z + d == 0:
In[]:=
​​Solve[2(4)+3(6)+4(8)+d==0,d]
Out[]=
{{d-58}}
The planes are:
2x+3y+4z==0;​​2x+3y+4z-58==0;
2) Circumscribed cylinder of parallel generatrix to x = y = z :
Method 1:
By the way, we answer section 3) Center and radius:
In[]:=
2
(x-2)
+
2
(y-3)
+
2
(z-4)
==29;​​C=={2,3,4};r=
29
;​​
29
==
Norm[{x-2,y-3,z-4}{1,1,1}]
Norm[{1,1,1}]
29

2
Abs[1+x-y]
+
2
Abs[1+y-z]
+
2
Abs[-2-x+z]
3
;
In[]:=
Simplify[Expand[29(3)==
2
(1+x-y)
+
2
(1+y-z)
+
2
(-2-x+z)
]]
2(3+
2
x
+
2
y
-3z-yz+
2
z
-x(-3+y+z))87;​​2
2
x
+2
2
y
+2
2
z
-2xy-2xz-2yz+6x-6z-81==0;
Method 2:
In[]:=
SimplifyExpand
2
(x
f
x
[1,1,1,0]+y
f
y
[1,1,1,0]+z
f
z
[1,1,1,0]+1
f
t
[1,1,1,0])
-4f[1,1,1,0]f[x,y,z,1]==0
2(
2
x
+
2
y
+
2
z
)81+2(3+y)z+2x(-3+y+z);​​2
2
x
+2
2
y
+2
2
z
-2xy-2xz-2yz+6x-6z-81==0;
Animation:
In[]:=
​​Manipulate[ζ[t_]:={2,3,4}+t{4,6,8};σ[u_,v_]:=
29
Cos[u]Sin[v]+2,
29
Sin[u]Sin[v]+3,
29
Cos[v]+4;​​i[x_,y_,z_]:=2
2
x
+2
2
y
+2
2
z
-2xy-2xz-2yz+6x-6z-81;​​g[x_,y_,z_]:=2x+3y+4z;​​h[x_,y_,z_]:=2x+3y+4z-58;P={0,0,0};Q={4,6,8};​​Show[{ParametricPlot3D[ζ[t],{t,-1,1},PlotStyleGray]/.Line[pts_,rest___]:>Tube[pts,0.08,rest],​​Graphics3D[{Black,Ball[P,0.15]}],Graphics3D[{Black,Ball[Q,0.15]}],ParametricPlot3D[σ[u,v],{u,0,2π},{v,0,π},PlotStyle->Opacity[0.6],MeshFalse,PerformanceGoal"Quality",ColorFunction"LightTemperatureMap"],ContourPlot3D[g[x,y,z]==0,{x,-10,10},{y,-10,10},{z,-15,15},MeshFalse,PerformanceGoal"Quality",ContourStyle->Directive[Purple,Opacity[0.2],Specularity[White,30]],MeshNone],ContourPlot3D[h[x,y,z]==0,{x,-10,10},{y,-5,15},{z,-60,60},MeshFalse,PerformanceGoal"Quality",ContourStyle->Directive[Purple,Opacity[0.2],Specularity[White,30]]],ContourPlot3D[i[x,y,z]==0,{x,-20,20},{y,-20,20},{z,-5,t1},MeshFalse,PerformanceGoal"Quality",ContourStyle->Directive[Yellow,Opacity[0.2],Specularity[White,30]],BoundaryStyleDirective[Black,Thick],PlotPoints120]},​​AxesLabel->(Style[#,15,Blue]&/@{"X","Y","Z"}),AxesOrigin{0,0,0},AxesTrue,BoxedFalse,BoxRatiosAutomatic,PlotRangeAll,ImageSize900,ViewPoint{-3,3,1}],Style["Sphere, circumscribed cylinder and tangent planes:",Bold,Large],{{t1,-4.9998,"Valor (t1)"},-4.9999,12,0.00001},ControlPlacementTop]
Out[]=
​
Sphere, circumscribed cylinder and tangent planes:
Valor (t1)