Do you have to play batman arkham asylum before arkham city

Not necessarily, I certainly would recommend playing Arkham Asylum first as their are pieces of the story that also tie into the sequel,but on the same token everything that happens inside of Arkham City is also stand alone so you could jump straight into the sequel without much confusion. Even if you are familiar with the Batman Mythos there are a few examples of brand new characters being introduced within Arkham Asylum.
All in All if you are debating about playing both simply play both. I'd recommend playing in chronological order but is playing them in order a necessity,no you can play in any order and not risk much confusion or much spoilers to the previous game other than its ending.