the beetle was crawling at a speed of 1 unit w.r.t. Let the centre of the circle be called C. Let the beetle be at point B at time t. Then the anle which CB makes with the x-axis is t radians. (do you see why?)
the position of the beetle is hence (10t+cos(t), sin(t)). eleminate t and you will get the trajectory.
EDIT: for the position of the beetle when it reaches the rear bumper use the equation (10t + cos(t),sin(t))=(10t-1,0) which gives t= pi
so the position of the beetle is (10pi-1,0)