residually finite proof

Let $A$ be a polycyclic-by-finite group.
$\langle a \rangle, \langle b \rangle \in center(A)$.
Also, $\langle a \rangle / (N \cap \langle a \rangle)$ is finite.

how to show that $A/(N\cap \langle a \rangle)$ is residually finite???

Is polycyclic-by-finite group finitely generated? If yes, then the proof just walks out.

Can I claim that $\langle a \rangle$ is polycyclic and imply 3rd isomorpism theorem?

any other way to prove this problem?