In[]:=
ThreeQuadricsConical::usage="Show that if three quadrics pass through the same conic, the planes of the other three intersection conicals pass through the same line. References: Usunáriz Balanzategui, Ubaldo; Usunáriz Sala, Ignacio. Problemas de Geometría​; Personal.";
Three quadrics pass through the same conical (in red):
In[]:=
Show[Conical=
2
x
+
2
y
-1;ConicalPlane=z;Plane1=2x-5y+7z+3;Plane2=x-2y+5;Plane3=5x-13y+z-2;​​Quadric1=Conical+ConicalPlanePlane1;Quadric2=Conical+ConicalPlanePlane2;​​Quadric3=Conical+ConicalPlanePlane3;{ContourPlot3D[{Quadric1==0,Quadric2==0,Quadric3==0},{x,-10,10},{y,-10,10},{z,-10,10},ContourStyle->Opacity[0.5],PlotPoints->60,MeshFalse],ParametricPlot3D[{Cos[θ],Sin[θ],0},{θ,0,2π},PlotStyle->{Red,Thick}]}]
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Intersection conicals on each of its planes:
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Plane12=Plane1-Plane2;Plane13=Plane1-Plane3;​​Plane23=Plane2-Plane3;
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​
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ShowParametricPlot3D-
Cos[t]-3Cos[t]+11Sin[t]+
23-86Cos[2t]-63Sin[2t]

7+5Cos[2t]+5Sin[2t]
,-
Sin[t]-3Cos[t]+11Sin[t]+
23-86Cos[2t]-63Sin[2t]

7+5Cos[2t]+5Sin[2t]
,
1
7
2+
Cos[t]-3Cos[t]+11Sin[t]+
23-86Cos[2t]-63Sin[2t]

7+5Cos[2t]+5Sin[2t]
-
3Sin[t]-3Cos[t]+11Sin[t]+
23-86Cos[2t]-63Sin[2t]

7+5Cos[2t]+5Sin[2t]
,{t,0,2π},PlotStyle{Black,Thick},ContourPlot3D[Plane12==0,{x,-10,20},{y,-50,20},{z,-25,10},MeshFalse,ContourStyle{Green,Opacity[0.4],Specularity[White,40]}]
Out[]=
In[]:=
ShowParametricPlot3D
Cos[t]-216Cos[t]+566Sin[t]+6
2759-2198Cos[2t]-1975Sin[2t]

-859+589Cos[2t]+522Sin[2t]
,
Sin[t]-216Cos[t]+566Sin[t]+6
2759-2198Cos[2t]-1975Sin[2t]

-859+589Cos[2t]+522Sin[2t]
,
1
6
-5+
3Cos[t]-216Cos[t]+566Sin[t]+6
2759-2198Cos[2t]-1975Sin[2t]

-859+589Cos[2t]+522Sin[2t]
-
8Sin[t]-216Cos[t]+566Sin[t]+6
2759-2198Cos[2t]-1975Sin[2t]

-859+589Cos[2t]+522Sin[2t]
,{t,0,2π},PlotStyle{Black,Thick},ContourPlot3D[Plane13==0,{x,-10,20},{y,-50,20},{z,-25,10},MeshFalse,ContourStyle{Green,Opacity[0.4],Specularity[White,40]}]
Out[]=
In[]:=
ShowParametricPlot3D
Cos[t]13Cos[t]-41Sin[t]+
2557-1980Cos[2t]-1825Sin[2t]

-24+18Cos[2t]+19Sin[2t]
,
Sin[t]13Cos[t]-41Sin[t]+
2557-1980Cos[2t]-1825Sin[2t]

-24+18Cos[2t]+19Sin[2t]
,7-
4Cos[t]13Cos[t]-41Sin[t]+
2557-1980Cos[2t]-1825Sin[2t]

-24+18Cos[2t]+19Sin[2t]
+
11Sin[t]13Cos[t]-41Sin[t]+
2557-1980Cos[2t]-1825Sin[2t]

-24+18Cos[2t]+19Sin[2t]
,{t,0,2π},PlotStyle{Black,Thick},ContourPlot3D[Plane23==0,{x,-10,20},{y,-50,20},{z,-50,60},MeshFalse,ContourStyle{Green,Opacity[0.4],Specularity[White,40]}]
Out[]=
The intersection planes and their common line:
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Solve[{Plane12==0,Plane13==0,Plane23==0},{x,y}]
Out[]=
{{x-1+74z,y-1+27z}}
In[]:=
{{x-1+74z,y-1+27z}};​​{-1+74t,-1+27t,t};