For what it's worth, I think this is what I wanted to provide before:

aa = Sqrt[13/3];
bb = Sqrt[13/3];
hyperbolaX[a_][t_] := a Sec[t];
hyperbolaY[b_][t_] := b Tan[t];Manipulate[
 Show[{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]}, {Directive[Black, 
        Dotted]}}], 
   ParametricPlot[(TranslationTransform[{-3/2, 7/(2 Sqrt[3])}]@*
       RotationTransform[-Pi/6])[{hyperbolaX[aa][t], 
      hyperbolaY[bb][t]}], {t, Pi/2 + .1, s}, 
    PlotStyle -> {Red, Thickness[.02]}]}, 
  AspectRatio -> Automatic], {{s, Pi}, Pi/2 + .2, 3 Pi/2 - .1}]

Maybe something like this:

Manipulate[
 Show[
  {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]}, {Directive[Black, 
        Dotted]}}],
   ParametricPlot[(TranslationTransform[{-3/2, 7/(2 Sqrt[3])}]@*
       RotationTransform[-Pi/6])[{hyperbolaX[Sqrt[a2]][t], 
      hyperbolaY[Sqrt[a2]][t]}], {t, Pi/2 + .1, s}, 
    PlotStyle -> {Red, Thickness[.02]}]}, AspectRatio -> Automatic],
 {{s, Pi}, Pi/2 + .2, 3 Pi/2 - .1}]

Where:

aa = Sqrt[13/3];
bb = Sqrt[13/3];
hyperbolaX[a_][t_] := a Sec[t];
hyperbolaY[b_][t_] := b Tan[t];

The hyperbolaX and hyperbolaY are the parameterization functions for a hyperbola (in standard orientation). Inside the Manipulate I'm rotating and translating to bring it into alignment with your "background" hyperbola.

You might try RegionFunction. Let's create a function to use:

MyRegionFunction[p_] := #2 > #1 Sqrt[3] - p &

Here's an updated version of your Manipulate. I removed some options and removed a stray ImageSize option, so you might need to put some things back in. The important change is with how sliderHyperbola is defined. There might be a more performant way to do this, but this at least gets you the functionality you want (I think :) ).

Manipulate[
 Module[
  {hyperbola, sliderHyperbola},
  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]}, {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]];
  Show[
   hyperbola,
   sliderHyperbola,
   Frame -> True,
   ImageSize -> 20.1 16.1,
   AspectRatio -> Automatic, PlotRange -> {{-9, 7}, {-6, 10}}]],
 {p, -14, 8}]

I get the output as expected, no errors, in both versions 12.0 and 13.2. What version are you using? Try with a fresh kernel.

I wonder why you set ImageSize -> 20.1 16.1, which seems bizarre to me. Did you mean to give separate width and height? That is done with ImageSize -> {20.1, 16.1}.

Don't worry about the name.

I'm not actually sure that I can do that. As far as I can tell, the RegionFunction option expects to be given something with head Function. So, I think we're going to end up with some "pure" function somewhere along the way. Sorry.

Hmm. That's weird. Maybe different version behave differently. I'm using 13.2. Anyway, glad you got something working!

I intended the ratio to be 16 by 16. But I'm also using diagonal gridlines which I left out of the programs I sent in order to simplify them. When the gridlines were in, even with Frame -> True, the right edge of the frame did not appear. As for the 16.1, I was confused. I thought I was making the frame wider but but the right edge was not appearing. Eventually I realized my mistake and changed 20 to 20.1, which allowed the right edge to show up. I forgot to change 16.1 back to 16.
As for what is wrong here, so far I've repasted it, but without success. I will keep testing. I'm using windows 10 from 2019 and I installed V13.2 when it came out.

All I can say is that your assistance is super remakable. I consider myself very lucky.

That post was mine (this is Eric). Sometimes this site sets the name to "Updating Name" for some reason. Must be some sort of error condition that triggers that.

Anyway, I would try to show you another way to write MyRegionFunction, but it's being used in the RegionFunction option, and that option seems to expect a Function expression, so there's no point in trying to show you a different way to do it.

Now, if your confusion is just with the # and & stuff, then another user (who's name I don't know because for me it's just showing as "Updating Name") wrote it this way: RegionFunction -> Function[{x, y}, y > x Sqrt[3] - p]]

If that's what you're asking for, then there you have it with thanks to that mystery, anonymous user.

I'm surprised and grateful that you are so attentive to questions like mine from the community! Your previous coding has been monumental to me in advancing to my latest Manipulate, even if it is not apparent.

However, the slider in your manipulate reveals no hyperbola.

Where I'm really confused is with your definitions for aa and bb. These terms, hence their values, are not used in the manipulate program. Also, in manipulate you have an undefined a2. Shouldn't a2 have a value or range of values in manipulate?

I did replace a2 with your definitions in all the ways I could think of, and I did get the slider to trace out a left branch of an approximate hyperbola from bottom to top. What I want, however is for the slider to reveal the red hyperbola so that what is shown is always parallel to the transverse axis. I hope the two screenshots shown below for two different slider positions will show what I mean.

Is it possible to reveal the red hyperbola as the slider moves the tilted (imaginary) red line back and forth along the main axis?

If my remarks seem like criticism, please know that that any code you write personalized for me is the best learning experience I have going. I think my failing to precisely communicate what I want while trying to do it in as few words as possible is a standard in this community that I'm just starting to learn. If I'm not getting what I want, it's a fault I'll have to overcome. I benefit from all your code. To have attracted your interest is almost beyond belief.