Consider the function

$$f(x, y)=\frac{x}{y}+\frac{y}{x}-\frac{(x-y)^{2}}{a^{2}}$$

defined for $x>0$ and $y>0$, where $a$ is a non-zero real constant. Show that $(\lambda, \lambda)$ is a stationary point of $f$ for each $\lambda>0$. Compute the Hessian and its eigenvalues at $(\lambda, \lambda)$.