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3 edited body
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 h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, g]z]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, k }}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

enter image description here

 h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, g]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, k }}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

enter image description here

 h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, z]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, k }}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

enter image description here

2 added 19 characters in body
source | link
 h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, h - g]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, -k }}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

ContourPlot3Denter image description here

h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, h - g]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, -k}}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

ContourPlot3D

 h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, g]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, k }}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

enter image description here

1
source | link

h = x^2 + y^2/9 + z^2/4 - 1;
g = z;
ContourPlot3D[{h == 0, g == 0, g == k}, {x, -1, 1}, {y, -3, 
  3}, {z, -2, 2}, MeshFunctions -> {Function[{x, y, z, f}, h - g]}, 
 MeshStyle -> {{Thick, Blue}}, Mesh -> {{0, -k}}, 
 ContourStyle -> 
  Directive[Orange, Opacity[0.5], Specularity[White, 30]]]

ContourPlot3D