How are the foci of a quadric defined?
By a quadric I mean a set $$ Q = \left\{ x \in \mathbb R^n \mid x^T A x + 2 b^T x + c = 0 \right\}, $$ where $A\in\mathbb R^{n\times n}$ is symmetric and $b\in\mathbb R^n$.
Edit: I ask this questions because I encountered the term "confocal quadrics". And I do not even know what a focus is in general dimensions. Maybe we can assume the most simple form $$Q=\left\{ x \in\mathbb R^n\mid \frac{x_1^2}{a_1}+\ldots+\frac{x_n^2}{a_n}=1 \right\},$$ where $a_1<\ldots <a_n$. What are the foci of this quadric?
