# Draw the plane on ParametricPlot3D with one parameter

I need to make semitransparent coordinate plane(s) at ParametricPlot3D with one parameter. If I add the second parameter the curves becomes ugly.

 GraphicsRow[{
ParametricPlot3D[{
{Sin[u], Tan[u], u/10},
0.5 {Cos[u], Sin[u], u/10}},
{u, -20, 20},  BoxRatios -> {1, 1, 1}],
ParametricPlot3D[{
{Sin[u], Tan[u], u/10},
0.5 {Cos[u], Sin[u], u/10},
{u/10, v/10, 0}},
{u, -20, 20}, {v, -20, 20}, BoxRatios -> {1, 1, 1}]}]


How I can fix it?

Just make two separate graphics and combine them using Show

Show[{ParametricPlot3D[{{Sin[u], Tan[u], u/10},
0.5 {Cos[u], Sin[u], u/10}}, {u, -20, 20}],
ParametricPlot3D[{u/10, v/10, 0}, {u, -20, 20}, {v, -20, 20},
PlotStyle -> {Opacity[0.3, Magenta]}]}, BoxRatios -> {1, 1, 1}]


In versions 10+, you can use InfinitePlane:

Show[ParametricPlot3D[{{Sin[u], Tan[u], u/10}, 0.5 {Cos[u], Sin[u], u/10}}, {u, -20, 20},
BoxRatios -> {1, 1, 1}],
Graphics3D@{Opacity[.3, Blue], InfinitePlane[{{0, 0, 0}, {0, 1, 0}, {1, 1, 0}}]}]