Cochran's theorem for correlated normals — only the whitened form is chi-square

If $(X,Y)'$ follows bivariate normal distribution with mean vector $(0,0)'$ and variance-covariance matrix $\Sigma=\left(\begin{smallmatrix}1&0.5\\0.5&1\end{smallmatrix}\right)$, which of the following does not have a $\chi^2$-distribution?