Question: tracing in animations question

I  have a little bit of code that chows a parabola "rolling" and its focus tracing out a catenary. I want to trac the focus but not the parabola, and am unsure how. Any suggestions? Here is my code bit:

> L := unapply(int(sqrt(1+(2*x)^2), x = 0 .. X), X);
> P1 := display([pointplot([0, 1/4]), plot(x^2, x = -2 .. 2)], color = blue);
> F := proc (t) options operator, arrow; rotate(translate(P1, L(t)-t, -t^2), -arctan(2*t), [L(t), 0]) end proc;
> animate(F, [t], t = -2 .. 2, trace = 10, scaling = constrained);
 

Any help appreciated.

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