Independent Standard Normal Gaussian Random Variables I am having trouble making sense of this. I know what independence of random variables means. Suppose $X$ and $Y$ are independent, standard normal (Gaussian) random variables. Then, it is supposed to be that $X^2 + Y^2$ and $\frac X Y$ are also independent random variables. 
I just cannot intuitively make sense of this. Where does that fact that they are standard normal Gaussian come into play? Also, if we know that $\frac X Y \leq k$ for some fixed $k$, then it seems obvious that it should affect the probability distribution of $X^2 +Y^2$. 

Additional remark to Byron's answer: It can actually be shown that an (isotropic) Normal distribution is the only rotationally symmetric distribution for which $X$ and $Y$ as well as $X/Y$ and $X^2 + Y^2$ are independent. 
See e.g. 

Ali, M. M. (1980). Characterization of the Normal Distribution Among
  the Continuous Symmetric Spherical Class. Journal of the Royal
  Statistical Society B, 42(2), 162-164.

