Describe briefly the method of Lagrangian multipliers for finding the stationary points of a function $f(x, y)$ subject to a constraint $g(x, y)=0$.

Use the method to find the stationary values of $x y$ subject to the constraint $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 .$