I may be overlooking something, but it seems to me it will be difficult to follow Mr. F's advice through to the conclusion of the problem. The difficulty is that since are not independent, we can't say that has a chi-square distribution.
Here is another way to proceed that avoids that difficulty. We know that have a joint bivariate normal distribution and that individually have (univariate) normal distributions with mean 0 and standard deviation .
Notice that
so are uncorrelated, hence independent.
Then
has a chi-square distribution with 2 degrees of freedom, and we can use this fact to compute
.
I was thinking of integrating the joint pdf for X1 and X2 over the annulus (possibly switching to polar coordinates to do so) but I hadn't really followed through the details.
But your solution is much, much better and is much more tractable. Thank you for showing it.
My only 'concern' is that a covariance of zero doesn't in general imply independence (although independence does imply a covariance of zero) ..... This is a small detail that can be left for the OP to worry about (s/he should know that there's a thereom about multivariate normal distributions, covariance and independence .....)