Add some details to cadjunkie's answer:

Assume the path of center is C(t) = (x(t), y(t)), line equation is L: ax+by=0, and the current time is t0.

"Translate" the line to find the intersection at t0, say it's P1(x1,y1).

Then at any time t, position of P1 would be P1(t) = C(t) + (x1-x(t0), y1-y(t0)) = (x1(t), y1(t))

Now solve the equation for t1:

a * x1(t1) + b * y1(t1) = 0

And t1 - t0 is the answer

Note we need special handling if line equation is y=0