Do you want to use the antenna for both the TV and the scanner at the same time?
Is the existing feedline 300 ohm twinlead (flat) or coaxial cable (round) ?
TV antennas are usually mounted with the antenna elements horizontal since broadcast TV is horizontally polarized. Most things you listen to on a scanner are vertically polarized and you would have better reception with the antenna elements mounted vertically (in other words, with the TV antenna rotated 90 degrees so the elements go up and down).
Luckily, most local police and fire departments have powerful repeaters and strong signals so most likely the TV antenna, even with the wrong polarization, will work for local signals.
My recommendation is to not worry too much about theory, and just hook the antenna up to the scanner and see if it works good enough to do what you want.
You will probably need to get some adaptors to go from the the BNC or SMA connector on your scanner to whatever sort of connector you have for your antenna system. For an initial attempt, I would not worry about the mismatch between a balanced 300 ohm twinlead and the unbalanced 50 or 75 ohm coaxial feed preferred by your scanner. It should only take a couple of minutes to connect the antenna to the scanner, even if only temporarily with bare wires stuck into connectors.