I have been working on a problem related to determinantal varieties in symmetric matrices. I am stuck at the following point and would like to get some reference/help for the following question. Let $\mathbb{F}_q$ be a finite field with odd characteristic and let $S(2t, m)$ be the set of all $m\times m$ symmetric matrices over $\mathbb{F}_q$ of rank $2t$ (even). For some $\delta\in\mathbb{F}_q$ a square (non-square) and $k\le m$ let $f^\delta_k(X)= X_{11}+\cdots+X_{k-1k-1}+\delta X_{kk}$. I want to know the cardinality of the following set $$ \{A\in S(2t, m): A\text{ is hyperbolic and }f^\delta_k(A)=0\}. $$ Here, by hyperbolic $A$ we mean that the corresponding quadric $XAX^T$ is hyperbolic. I am stuck at this point. My approach was to use some induction on $k$. To do so, I was thinking to project a symmetric matrix to an $m-1\times m-1$ matrix by deleting its first row and first column. But I have no control over the behavior of the fiber of this map. For example, if we take an $m-1\times m-1$ symmetric matrix of rank $2t-2$, $ 2t-1$ or $2t$ and add a new row and column to obtain a matrix in $S(2t, m)$, what are the odds to get a hyperbolic matrix?
I know this stuff is quite classical and probably this problem is already well understood. But unfortunately, I could not find references that only gives the number of symmetric matrices that are hyperbolic and elliptic.
