Solve Cycloid Question to Find Parameter 'a

[brainpushups]

Hi. I was working through a problem and got stuck. The problem asks "To find the value a for a given point (x2, y2) usually requires solution of a transcendental equation. Here are two cases where you can do it more simply: For x2 = (Pi)b, y2 = 2b and again for x2= 2(Pi) b and y2 = 0 find the value of a for which the cycloid goes through the point 2 and find the corresponding minimum times."

To clarify, the parametric equations for a cycloid are

x(t) = a (t - sin[t])
y(t) = a (1- cos[t])

and the question asks to find the constant 'a' for which the cycloid curve contains the point (x2, y2) and then find the time it takes the particle to reach the point.

I haven't been able to figure out how to find the parameter a. I have only attempted the solution for the first point. My thought was to simply eliminate the parameter t and then solve for a in terms of b, but this doesn't eliminate the transcendental nature of the equation. Does anybody have a suggestion?I also have an additional question. I understand that solutions to transcendental equations require graphical or numerical techniques, but I don't understand how that can be applied to this problem (if you were given a certain numeric ordered pair for (x2,y2) for instance). How would this be done?

 

[brainpushups]

Wow, nevermind. It is quite easy to see by inspection that a = b ;)