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?