I don't know that the line should be that different for power, but I have found that, at least on a race track, having bigger tires, better brakes, and more power allows you to get away with "other lines" without it being so noticeable to other people. Of course, if you take the right line, you should be able to get around quicker.
What I will say is that car setup can definitely affect your approach to a turn. If your car is loose (and more throttle-on torque in a rear-wheel-drive car can contribute to this feeling), you may tend to brake a little earlier (so you don't loop it), make sure you have some margin for the corner, and then accelerate (or coast) through the middle of the corner. If you car is tight (and you have ABS), it is possible to brake later, perhaps even enter the corner a little too fast, and still get away with it by scrubbing some speed off during corner entry (albeit with a somewhat funny line). The most obvious example is a front-wheel-drive car (usually tight) versus a rear-wheel-drive car (if it's an early 911, it's loose), and all-wheel-drive cars can also be a little different (push-me/pull-you). The approach to corner entry, the attitude of the car on exit, and the resulting lines tend to be different.