QJEEM25Ma4907: JEE Main 2025 Mathematics - Class 11 JEEM25Ma4907 Solution: Let integers be such that . Then the number of all possible…

$\begin{aligned}& a, b \in I,-3 \leq a, b \leq 3, a+b \neq 0 \\& |z-a|=|z+b| \\& \left|\begin{array}{ccc}z+1 & \omega & \omega^2 \\\omega & z+\omega^2 & 1 \\\omega^2 & 1 & z+\omega\end{array}\right|=1 \\& \Rightarrow\left|\begin{array}{ccc}z & z & z \\\omega & z+\omega^2 & 1 \\\omega^2 & 1 & z+\omega\end{array}\right|=1\end{aligned}$

$\begin{aligned}& \Rightarrow z\left|\begin{array}{ccc}1 & 1 & 1 \\\omega & z+\omega^2 & 1 \\\omega^2 & 1 & z+\omega\end{array}\right|=1 \\& \Rightarrow z\left|\begin{array}{ccc}1 & 0 & 0 \\\omega & z+\omega^2-\omega & 1-\omega \\\omega^2 & 1-\omega^2 & z+\omega-\omega^2\end{array}\right|=1\end{aligned}$

$\begin{aligned}&\begin{aligned}& \Rightarrow z^3=1 \\& \Rightarrow z=\omega, \omega^2, 1\end{aligned}\\&\text { Now }\\&\begin{aligned}& |1-\mathrm{a}|=|1+\mathrm{b}| \\& \Rightarrow 10 \text { pairs }\end{aligned}\end{aligned}$