  Let $(P_N)_{N\ge0}$ one of the classical sequences of orthogonal polynomials,
i.e., Hermite, Laguerre or Jacobi polynomials. For the roots $z_{1,N},\ldots,
z_{N,N}$ of $P_N$ we derive lower estimates for $\min_{i\ne
j}|z_{i,N}-z_{j,N}|$ and the distances from the boundary of the orthogonality
intervals. The proofs are based on recent results on the eigenvalues of the
covariance matrices in central limit theorems for associated $\beta$-random
matrix ensembles where these entities appear as entries, and where the
eigenvalues of these matrices are known.
