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I have a curve $\gamma=(f(x),g(x))$ parameterized by a real parameter $x > 1$ (one can perform a variable change $y = x^{-1} \in (0, 1)$ if $x\rightarrow\infty$ is problematic), with the functions $f(x)$ and $g(x)$ known. I wish to plot this in 2D using Mathematica, but I don't know how to do it.

In my case \begin{equation} f(x) = \frac{(1 - x^2)\cos(\theta)\sin(\theta)}{\cos^2(\theta) + \sin^2(\theta)x^2} \ , \quad g(x) = -\frac{x}{\cos^2(\theta) + \sin^2(\theta)x^2} \ , \end{equation}

where $\theta \in (0, \frac{\pi}{2}]$ (I'm thinking of using Manipulate to plot different values of $\theta$).

So far I have only managed to use ListPlot to draw different points for $x$ in 2D:

Manipulate[ListPlot[Table[{f[x], g[x]}, {x, 1, 10}]], {\[Theta], 0, \[Pi]/2}]

I would like to draw the entire curve and not just some points along it. Help is appreciated.

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    $\begingroup$ Have a look at Menu/WolframDocumentation/ParametricPlot. You can use Manipulate with it the same way you did in your attempt. $\endgroup$ – Alexei Boulbitch Mar 17 at 6:43
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ParametricPlot[{((1 - x^2)*Cos[θ] Sin[θ])/(Cos[θ]^2 + Sin[θ]^2*x^2), -(x/(Cos[θ]^2 + Sin[θ]^2*x^2))}, {θ, 0, π/2}, {x, 1, 10}, MeshFunctions -> {#3 &, #4 &}, 
 MeshStyle -> {Red, Cyan}, Mesh -> 20, 
 PlotStyle -> {Opacity[.1], Yellow}, PlotPoints -> 80]

enter image description here

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Here is a solution with Manipulate:

Manipulate[
 ParametricPlot[{((1 - x^2)*
      Cos[\[Theta]] Sin[\[Theta]])/(Cos[\[Theta]]^2 + 
      Sin[\[Theta]]^2*
       x^2), -(x/(Cos[\[Theta]]^2 + Sin[\[Theta]]^2*x^2))}, {x, 1, 
   50}, PlotRange -> {{-5, 0}, {-4, 0}}], {\[Theta], 0, \[Pi]/2}]
 

If we replace x -> 1/x we get:

Manipulate[
 ParametricPlot[{((1 - x^2)*
       Cos[\[Theta]] Sin[\[Theta]])/(Cos[\[Theta]]^2 + 
       Sin[\[Theta]]^2*
        x^2), -(x/(Cos[\[Theta]]^2 + Sin[\[Theta]]^2*x^2))} /. 
   x -> 1/x, {x, 0, 1}, PlotRange -> {{0, 5}, {-4, 0}}], {\[Theta], 
  0, \[Pi]/2}]

enter image description here

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