Let $A$ be a real $3 \times 3$ symmetric matrix with eigenvalues $\lambda_{1}>\lambda_{2}>\lambda_{3}>0$. Consider the surface $S$ in $\mathbb{R}^{3}$ given by

$$x^{T} A x=1$$

Find the minimum distance between the origin and $S$. How many points on $S$ realize this minimum distance? Justify your answer.