Symmetric Matrices & Quadratic Forms · Hard problem

The quadratic form $q(x,y,z) = 2x^2+2y^2-z^2$ is indefinite on all of $\mathbb R^3$ (its matrix has eigenvalues $2,2,-1$, mixed signs). But restricted to the 2-dimensional subspace $z=x$ (parametrized by $x,y$), it becomes $q = x^2+2y^2$. Find the SMALLER eigenvalue of this restricted form's $2\times2$ matrix.