ag.algebraic geometry – Symmetric matrices of Hyperbolic and elliptic type with certain kind of trace zero

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 $mtimes m$ symmetric matrices over $mathbb{F}_q$ of rank $2t$(even). For some $deltainmathbb{F}_q$ a square (non-square) and $kle m$ 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
$$
{Ain S(2t, m): Atext{ is hyperbolic and }f^delta_k(A=0}.
$$

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 $m-1times m-1$ 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 $m-1times m-1$ 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.