Depending on the TV they have varying native resolutions(meaning the actual number of pixels). Despite being marketed as a 1080p capable set that does not mean that it has 1920x1080 square pixels. Some TVs have fewer pixels in one direction and they are actually rectangular pixels. The hardware inside of the set does some stretching on the image before sending it out and the final displayed result looks like a perfectly proportioned image but it would be blurry for text.
In order to get the best picture possible you would need to find out what the native resolution actually is and have the PC output that resolution, color depth, and refresh rate. This causes the minimum amount of processing to be done on the image before being displayed. You'd also need to make sure that the TV isn't in any fit/stretch/zoom mode. But bear in mind that if you are looking at a 40" screen, it is going to be nearly 4X the real estate of most desktop monitors but have the same number of pixels. It's going to be blown up and so there may be some post-processing that is done to make the image look better (ie blurring).
Graphics cards have no inherent aspect ratio so your question about 16:9 vs 16:10 isn't relevant. If you have a 16:10 display then you just choose a resolution that has a 16:10 aspect ratio and likewise for 16:9. Now, if the PC was set to a resolution that was a different aspect ratio than the TV it might cause black bars but that is totally fixable.
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