# How can I draw the intersection of a cone and a plane?

I have a cone and a plane myQ = x - 4 y + z + 4 == 0;. How can I draw the intersection of the cone and the plane myQ?

  Clear["Global*"]
pA = {2, 3, 0};
myP = 2 x + y + 2 z - 1 == 0;
reg1 = ImplicitRegion[x \[Minus] 4  y + z + 4 == 0, {x, y, z}];
pH = RegionNearest[ImplicitRegion[myP, {x, y, z}]]@pA;
d = Sqrt[5^2 - EuclideanDistance[pA, pH]];
reg2 = Cone[{pA, pH}, d];
r2 = RegionIntersection[reg1, reg2];
Graphics3D[{Gray, Opacity[0.5], reg2}, Boxed -> False]

Show[Region[Style[r2, Cyan]]]


Show[Region[Style[reg1, Red]], Region[Style[reg2, Blue]]]


Set PlotRange in Region.

Show[Region[Style[reg1, Opacity[.3], Red], PlotRange -> 8],
Region[Style[reg2, Opacity[.2], Blue]], Region[Style[r2, Cyan]]]


• Add BoundaryStyle.
Show[Region[Style[reg1, Opacity[.3], Red], PlotRange -> 8],
Region[Style[reg2, Opacity[.2], Blue]],
RegionPlot3D[DiscretizeRegion@r2,
BoundaryStyle -> Directive@{Dashed, Thick}, PlotStyle -> Cyan]]
`

• Can I get the equation of the parabol is intersection of the cone and the plane? Mar 23 at 3:07