I'm trying to figure out where the interecting points are in the confidence region given by Hotteling's T^2 statistic.

I know the confidence ellipsoid is given by

+/- \sqrt{\frac{\lambda_i}{n}}\sqrt{\frac{(n-1)p}{n-p} F_{p,n-p} and centered at \overline{x}.

Would the points of intersection be \sqrt{\lambda_i} or \sqrt{\lambda_i / n}?

Any help would be appreciated.