  Given a quadratic function $f(x)=x^TAx+2a^Tx+a_0,$ it is possible that its
level set $\{x\in\mathbb{R}^n: f(x)=0\}$ has two connected components and thus
can be separated by the level set $\{x\in\mathbb{R}^n: g(x)=0\}$ of another
quadratic function $g(x)=x^TBx+2b^Tx+b_0.$ It turns out that the separation
property of such kind has great implication in quadratic optimization problems
and thus deserves careful studies. In this paper, we characterize the
separation property analytically by necessary and sufficient conditions as a
new tool to solving optimization problems.
