Check this out FIRST before you spend any money:
https://support.apple.com/en-us/HT203787
iMac (27-inch): AMD Radeon 6970M Video Card Replacement Program
Apple has determined that some AMD Radeon HD 6970M video cards used in 27-inch iMac computers with 3.1GHz quad-core Intel Core i5 or 3.4GHz quad-core Intel Core i7 processors may fail, causing the computer’s display to appear distorted, white or blue with vertical lines, or to turn black. iMac computers with affected video cards were sold between May 2011 and October 2012.