This is quite a detailed probability theory question and any hints or a place to start would be greatly appreciated. The question is as follows:
A radioactive source emits particles with all directions being equally likely. The source is held at distance 1 from a vertical infinite plane photographic plate. Consider the nearest point of that plane to the source as the origin. Show that, given the particle hits the plate, the horizontal coordinate of the point of its impact has the Cauchy distribution, i.e., the density function is $$f_X(x)= \frac{1}{π(x^2+1)}$$