Part of what is going on is Windows has different power settings for battery versus a/c operation. In Control Panel, find the Power Options section. When plugged in, it's possible that you are running on High Performance, which means your processor ramps up for every little thing you do. That is probably why that corner of the laptop gets hot.

Play around with one of the power plans to find one that makes sure you aren't running a fan all the time, but still gives you reasonable performance.