Eigenvalues · Hard problem

A force applied in the direction $\begin{pmatrix}1\\3\end{pmatrix}$ to a structure causes a response that grows without bound (resonance) exactly when that direction is an eigenvector of the response matrix $A = \begin{pmatrix}2 & k\\6 & 5\end{pmatrix}$. Find the value of $k$ for which resonance occurs.