Distance extrema on a central plane section of an ellipsoid ($d_{\min }^2={\lambda }_{\max }(U^{T}AU{)}^{-1},d_{\max }^2={\lambda }_{\min }(U^{T}AU{)}^{-1}$)

Let $A$ be a real positive-definite matrix and let the columns of $U$ be an orthonormal basis of a two-dimensional linear subspace in three-dimensional Euclidean space. Points in that central plane are $x=Uq$, with $|x{|}^2=|q{|}^2$. Restricting the ellipsoid $x^{T}Ax=1$ gives $q^T(U^{T}AU)q=1$. The compressed matrix $U^{T}AU$ is a positive-definite matrix, and its extreme eigenvalues give the distance formulas in the title by the Rayleigh quotient. The two extremal axes are the images under $U$ of the corresponding eigenvectors. This restriction reduces a problem with two Lagrange multipliers to a two-dimensional quadratic form.