If there were ever such a thing as a Mathematica "high," I'm in one today, thanks to you, Eric. It was just before Christmas when I had my doubts that it was even possible to get a line to move parallel to itself.
As for your last program with the infinite line, I kept getting an error on execution, even after quitting the kernel. This was the same error I was getting yesterday. I did not have the success reported by Gianluca Gorni.
Eventually it occurred to me that the defined variables were not enclosed in Module. Therefore, I took your new code for the infinite line, and put it in the program that had the Module. I'm writing a paper and I wanted to use this Manipulate to illustrate it. Without your help I never would have developed the Manipulate. About all I did was furnish the idea.
Thank you for bringing into existence what I could only imagine. Here is how I put the variables inside Module:
Manipulate[
Module[{hyperbola, sliderHyperbola, sliderLine},
hyperbola =
ContourPlot[{-10 x - x^2 + (2 y)/Sqrt[3] + 2 Sqrt[3] x y + y^2 ==
0, y == 2/Sqrt[3] - x/Sqrt[3],
y == Sqrt[3] x + 8/Sqrt[3]}, {x, -14, 9}, {y, -6, 10},
ContourStyle -> {{Directive[Black, Dotted]}}];
sliderHyperbola =
ContourPlot[-10 x - x^2 + (2 y)/Sqrt[3] + 2 Sqrt[3] x y + y^2 ==
0, {x, -14, 9}, {y, -6, 10}, ContourStyle -> Directive[Red],
RegionFunction -> MyRegionFunction[p]];
sliderLine = Graphics[{Blue, InfiniteLine[{0, -p}, {1, Sqrt[3]}]}];
Show[hyperbola, sliderHyperbola, sliderLine, Frame -> True,
ImageSize -> 400.1, AspectRatio -> Automatic,
PlotRange -> {{-9.5, 6.5}, {-6, 10}}]], {p, -28, 19, 0.025},
ContinuousAction -> True]