This is a homework problem. Let $(X,Y)\sim N(\mu_1,\mu_2,\sigma^2_1,\sigma^2_2,\rho)$. Show that if $\sigma_1,\sigma_2 >0,|\rho|<1$, then $$ \dfrac{1}{1-\rho^2}\left\{\dfrac{(X-\mu_1)^2}{\sigma^2_1}-2\rho\dfrac{(X-\mu_1)(Y-\mu_2)}{\sigma_1\sigma_2}+\dfrac{(Y-\mu_2)^2}{\sigma^2_2}\right\}$$ has a $\chi^2_2$ distribution.
EDIT:
There is a generalization of this: $(X-\mu)^T\Sigma^{-1}(X-\mu) \sim \chi^2_n$ where $X=(X_1,\ldots,X_n)^T$. The proof is theorem 7 here (http://www2.econ.iastate.edu/classes/econ671/Hallam/documents/QUAD_NORM.pdf)
Can anyone think of a better way to show this?