The GPU in the 27 inch screen model iMacs isn't a video card, but a socketed chip.
If the GPU has failed and If you know what VRAM GPU was in your iMac, you can ask if the faster, optional GPU that was available for that model year can be had and that you'll pay the extra costs to upgrade it.
The base GPU for the 2010, 27 inch screen iMac models was a 512 MB VRAM GPU.
The optional upgraded GPU for that model year was an ATI Radeon HD 5750 with 1 GB of dedicated GDDR5 memory.
IF your original GPU failed and if the GPU was the base 512 MB VRAM GPU, and if Apple still has these ATI HD5750 1 GB GPUs still available, you can ask the repair center to order the better GPU and tell them you'll pay for this GPU upgrade replacement.