# Plotting cross-sections of $z=4x^2+y^2$ parallel to the $yz$-plane

I want to plot the cross-sections of $z=4x^2+y^2$ parallel to the $yz$-plane. I have tried the below code

ContourPlot3D[x, {x, -6, 6}, {y, -6, 6}, {z, -1, 40},
Contours -> {-6, -4, -2, 0, 2, 4, 6}, ContourStyle -> {Opacity[0.3]},
PlotPoints -> 30, MaxRecursion -> 3, Mesh -> {{0}},
MeshFunctions -> {Function[{x, y, z}, z - (4 x^2 + y^2)]},
MeshStyle -> Directive[Thick, Red], AxesLabel -> Automatic]


and here is the output I get

It seems that I do not get any parabolas for $x=\pm4,\pm6$. How do I fix this?

• Either adjust the PlotRange in your method, or just use Plot3D[] directly: Plot3D[4 x^2 + y^2, {x, -6, 6}, {y, -6, 6}, BoundaryStyle -> None, BoxRatios -> {1, 1, 3}, Mesh -> 10, MeshFunctions -> {#1 &}, MeshStyle -> Red, PlotStyle -> None]. – J. M. is in limbo Mar 27 '18 at 8:31

## 1 Answer

A simple approach by using Show:

par = Plot3D[4 x^2 + y^2, {x, -10, 10}, {y, -10, 10},
BoundaryStyle -> None, BoxRatios -> {1, 1, 2}, PlotStyle -> Cyan, Mesh -> None];

sec = ContourPlot3D[x, {x, -10, 10}, {y, -10, 10}, {z, 0, 500},
Contours -> {-6, -4, -2, 0, 2, 4, 6},
ContourStyle -> {Opacity[0.3]}, PlotPoints -> 30, MaxRecursion -> 3,
Mesh -> {{0}}, BoxRatios -> {1, 1, 2},
MeshFunctions -> {Function[{x, y, z}, z - (4 x^2 + y^2)]},
MeshStyle -> Directive[Thickness[0.02], Red], AxesLabel -> Automatic];

Show[{par, sec}, Boxed->False]