Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b}
eq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}} ight|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \ \omega & z+\omega^2 & 1 \…

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Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} 
eq 0$. Then the number of all possible ordered pairs (a, b), for which $\left|\frac{z-\mathrm{a}}{z+\mathrm{b}}ight|=1$ and $\left|\begin{array}{ccc}z+1 & \omega & \omega^2 \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega\end{array}ight|=1, z \in \mathrm{C}$, where $\omega$ and $\omega^2$ are the roots of $x^2+x+1=0$, is equal to ________.

Aicharya · Accepted Answer

$\begin{aligned} & a, b \in I,-3 \leq a, b \leq 3, a+b 
eq 0 \ & |z-a|=|z+b| \ & \left|\begin{array}{ccc}z+1 & \omega & \omega^2 \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega\end{array}ight|=1 \ & \Rightarrow\left|\begin{array}{ccc}z & z & z \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega\end{array}ight|=1\end{aligned}$ $\begin{aligned} & \Rightarrow z\left|\begin{array}{ccc}1 & 1 & 1 \ \omega & z+\omega^2 & 1 \ \omega^2 & 1 & z+\omega\end{array}ight|=1 \ & \Rightarrow z\left|\begin{array}{ccc}1 & 0 & 0 \ \omega & z+\omega^2-\omega & 1-\omega \ \omega^2 & 1-\omega^2 & z+\omega-\omega^2\end{array}ight|=1 \ & \Rightarrow z^3=1 \ & \Rightarrow z=\omega, \omega^2, 1\end{aligned}$ Now $\begin{aligned} & |1-\mathrm{a}|=|1+\mathrm{b}| \ & \Rightarrow 10 	ext { pairs } \end{aligned}$

Let integers $\mathrm{a}, \mathrm{b} \in[-3,3]$ be such that $\mathrm{a}+\mathrm{b} \neq 0$. Then the number…

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