# How to display Parametric plots in different regions together?

I cannot find how to plot the branches of a 'ParametricPlot' in different colors

x[a_] := 8*(6 + a)/(16 - a^2);
y[a_] := 2*(8 + 3 a)/(16 - a^2);
pl1 = ParametricPlot[{x[a], y[a]}, {a, -40, -4}, PlotStyle -> Red]
pl2 = ParametricPlot[{x[a], y[a]}, {a, -4, 4}, PlotStyle -> Green]
pl3 = ParametricPlot[{x[a], y[a]}, {a, 4, 12}, PlotStyle -> Black]
Show[pl1, pl2, pl3]


Set Mesh->{{-4,4}}.( The default MeshFunctions->{#3&})

x[a_] = 8*(6 + a)/(16 - a^2);
y[a_] = 2*(8 + 3 a)/(16 - a^2);
ParametricPlot[{x[a], y[a]}, {a, -40, 12},
Mesh -> {{-4, 4}},PlotRange -> 10,(* MeshFunctions->{#3&}*)]



Add the PlotRange option to your Show statement, like this:

Show[{pl1, pl2, pl3}, PlotRange -> {{-8, 8}, {-5, 5}}]


We might ask what happens at $$a=-4$$? Why don't the three lines intersect at this point? Use Manipulate to see why:

Manipulate[
Show[{pl1, pl2 , pl3,
Graphics[{PointSize[.02], Cyan, Point[{x[a], y[a]}]}]},
PlotRange -> {{-8, 8}, {-5, 5}}],
{a, -10, 10}]

• Indeed the manipulate helps understand parametric curves with several branches, Thanks! Nov 28 '21 at 18:57