Mathematics Question from JEE Main 2025 · 29 January, Shift 2

Let integers $a, b \in [-3,3]$ be such that $a + b \neq 0$.

Then the number of all possible ordered pairs $(a, b)$, for which

$$\left| \frac{z - a}{z + b} \right| = 1$$

and

$$\begin{vmatrix} z+1 & \omega & \omega^2 \\ \omega & z+\omega^2 & 1 \\ \omega^2 & 1 & z+\omega \end{vmatrix} = 1, \; z \in \mathbb{C},$$

where $\omega$ and $\omega^2$ are the roots of $x^2 + x + 1 = 0$, is equal to: