Yeah, you can generally "whiten" the problem by scaling it in each axis until the variance is the same in each dimension. What you describe is if x and y have a covariance matrix of like

    [ σ², 0;
      0,  (nσ)² ]

but whitening also works in general for any arbitrary covariance matrix too.

[1] https://en.wikipedia.org/wiki/Whitening_transformation