Q. how do you find out if the movie rights to a book have been purchased?
You can contact the book publisher and ask them about who owns the film rights. If you can not find the publisher or the owner of the book's copyright you can try searching the records of the U.S. Copyright Office. Once you find out who owns the copyright contact them to find out about the film rights. Am including the link to the U.S. Copyright public catalog below.
If the copyright to the book has expired you can check to see if it's already in the public domain. The U.S. copyright office can help you with that as well. Visit their website or give them a call, 1-877-476-0778. Although they will charge you a fee if they do any lengthy searching for public domain information regarding individual book titles.