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 conic (in black):
In[]:=
ShowContourPlot3D
2
x
16
+
2
y
9
+
2
z
4
==1,
2
x
16
+
2
y
9
-
2
z
4
==1,
2
x
16
-
2
y
9
-
2
z
4
==1,{x,-6,6},{y,-4,4},{z,-3,3},ParametricPlot3D[{4Cos[θ],3Sin[θ],0},{θ,0,2π},PlotStyle->{Black,Thick}]
Out[]=
​
The other three intersection conicals:
In[]:=
Solve
2
x
16
+
2
y
9
+
2
z
4
==1,
2
x
16
+
2
y
9
-
2
z
4
==1,{x,z}
x-
4
3
9-
2
y
,z0,x
4
9-
2
y
3
,z0;±
4
3
9-
2
t
,t,0
In[]:=
Solve
2
x
16
+
2
y
9
-
2
z
4
==1,
2
x
16
-
2
y
9
-
2
z
4
==1,{y,z}
y0,z-
1
2
-16+
2
x
,y0,z
1
2
-16+
2
x
;t,0,±
1
2
-16+
2
t

In[]:=
Solve
2
x
16
+
2
y
9
+
2
z
4
==1,
2
x
16
-
2
y
9
-
2
z
4
==1,{x,y}
x-4,y-
3z
2
,x-4,y
3z
2
,x4,y-
3z
2
,x4,y
3z
2
;±4,±
3t
2
,0
The planes of the other three intersection conicals:
z==0;y==0;
In[]:=
Show[{ContourPlot3D[{z==0,y==0},{x,-4,4},{y,-4,4},{z,-4,4},ContourStyleOpacity[0.3]],ParametricPlot3D[t{1,0,0},{t,-6,6},PlotStyle{Black,Thick}]}]
Out[]=
The intersection line is t{1, 0, 0} the X-axis.
This checks the original statement.