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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):
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Show[Conical=
2
x
-
2
y
-1;ConicalPlane=z;Plane1=6x-5y+7z-3;Plane2=x+y;Plane3=3x-2y+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[{Cosh[θ],Sinh[θ],0},{θ,-π,π},PlotStyle->{Red,Thick}],ParametricPlot3D[{-Cosh[θ],Sinh[θ],0},{θ,-π,π},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]+3Sin[t]+
23+42Cos[2t]+23Sin[2t]

1+3Cos[2t]+Sin[2t]
,-
Sin[t]3Cos[t]+3Sin[t]+
23+42Cos[2t]+23Sin[2t]

1+3Cos[2t]+Sin[2t]
,
1
7
3+
5Cos[t]3Cos[t]+3Sin[t]+
23+42Cos[2t]+23Sin[2t]

1+3Cos[2t]+Sin[2t]
-
6Sin[t]3Cos[t]+3Sin[t]+
23+42Cos[2t]+23Sin[2t]

1+3Cos[2t]+Sin[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]}]
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ShowParametricPlot3D
Cos[t]-16Cos[t]+14Sin[t]+
-526+594Cos[2t]+528Sin[2t]

6(-4+3Cos[2t]+4Sin[2t])
,
Sin[t]-16Cos[t]+14Sin[t]+
-526+594Cos[2t]+528Sin[2t]

6(-4+3Cos[2t]+4Sin[2t])
,
1
6
1-
Cos[t]-16Cos[t]+14Sin[t]+
-526+594Cos[2t]+528Sin[2t]

2(-4+3Cos[2t]+4Sin[2t])
+
Sin[t]-16Cos[t]+14Sin[t]+
-526+594Cos[2t]+528Sin[2t]

2(-4+3Cos[2t]+4Sin[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]}]
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ShowParametricPlot3D
Cos[t]-2Cos[t]-2Sin[t]+
6
1-Cos[2t]+Sin[2t]

1-3Cos[2t]+Sin[2t]
,
Sin[t]-2Cos[t]-2Sin[t]+
6
1-Cos[2t]+Sin[2t]

1-3Cos[2t]+Sin[2t]
,2-
2Cos[t]-2Cos[t]-2Sin[t]+
6
1-Cos[2t]+Sin[2t]

1-3Cos[2t]+Sin[2t]
+
3Sin[t]-2Cos[t]-2Sin[t]+
6
1-Cos[2t]+Sin[2t]

1-3Cos[2t]+Sin[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]}]
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The intersection planes and their common line: