Yes you can be put on birth control pills to relieve cysts if you are as long as you are still within age to ovulate. I know of many people that take bcps to control cyst growth. Good luck!
Generally I don't think they give pills to resolve a cyst. Many cysts will resolve on their own. If it doesn't, they may want to do a laparascopy (small incision surgery) to remove it before it twists around on itself or ruptures (both of which are really painful).
Did your dr recommend BC pills? I could be wrong, maybe they do help if someone has cysts that keep coming back.