"Never-returning billiard balls"

Here's a problem I'm strugling with dealing with bouncing billiard balls.

A ball is placed in a position P inside a rectangular billiard of size a x b (a < b).

Here we discount any kind of friction and using standard reflection rule:

a) Show that there exists at least one direction such that if the ball starts along such direction and continue bouncing from the walls of the billiard, then it will never return to its initial position.

b) Let Q be any point inside the billiard. Find one direction such that the ball goes from P to Q, hitting the boundary just once; is this direction unique?

c) Find one direction such that the ball goes from P to Q, hitting the boundary twice; is this direction unique?

d) Generalize the results of b) and c) to the case of *n* bounces (*n* is a positive integer)

However I'm not sure wether it's best to solve this problem by euclidian geometry or some kind of analytical geometry (or any another method).

Thank you very much for help!