In[]:=
Manipulate[​​Show[​​{Graphics3D[{Opacity[0.8],Cone[{{0,0,-2.001},{0,0,0}},2.001],​​Cone[{{0,0,2.001},{0,0,0}},2.001]},​​PlotRange{{-2,2},{-2,2},{-2,2}},AxesTrue],​​ContourPlot3D[-a*x+0y+(1-Abs[a])*z-d==0,{x,-2,​​2},{y,-2,2},{z,-2,2},BoxedFalse,​​BoundaryStyleNone,ContourStyleOpacity[0.7,Yellow],​​MeshFunctions​​Function[{x,y,​​z},((x-0)^2+(y-0)^2-((z-0)*1)^2)-(-a*x+​​0y+(1-Abs[a])*z-d)],​​MeshStyle{{Thick,Darker[Green,0.5]}},Mesh{{0}},​​PlotRange{{-2,2},{-2,2},{-2,2}},MaxRecursion10]},​​Boxedboxed,Axesosy,ImageSize{400,400},​​SphericalRegionTrue],​​"plane shift",​​{{d,0,""},-2,2,0.02,Appearance"Labeled",ImageSizeTiny},​​"",​​"plane slope",​​{{a,0,""},-1,1,0.02,Appearance"Labeled",​​ImageSizeTiny},​​"","",​​"display axes",​​{{osy,True,""},{True,False}},​​"",​​"display the bounding box",​​{{boxed,False,""},{True,False}},​​​​Bookmarks{"degenerate conic: a point"{a=0.,d=0},​​"degenerate conic: a straight line"{a=-0.5,d=0},​​"degenerate conic: a pair of intersecting lines"{a=-0.68,​​d=0},"elliptical cross-section"{a=-0.16,​​d=0.66},​​"hyperbolic cross-section"{a=-0.78,d=0.52},​​"parabolic cross-section"{a=-0.5,d=0.52}},​​SaveDefinitionsTrue]
Out[]=
​
plane shift
0.5
plane slope
0.68
display axes
display the bounding box