## Hotteling's Confidence Region

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.