Is your "LED Strip", an array of phototransister(or LDR)-LED pairs similar to this?http://www.pololu.com/catalog/product/961
If so, you want to mount the array so that the line intersecting all the LEDs is perpendicular to the line you are trying to follow. Your path only needs to be wide enough to make at least one of the sensors out of the 8 register a change in reflected light.
The reason the array is so wide is so your robot can know not only if it is on the line or not, but also how far off of center it is. You can build a line-follower that has only a single sensor instead of the eight used in an array like this but it won't be able to go very fast. By increasing the number of sensors you increase the resolution at which you can determine how much you need to correct to get back over the line which leads to smoother turns which leads to higher speeds.
The higher speeds is the fun part.
So most like this I guess:
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