This looks like a very innocent question, but it is not.

Let's look at just one such point.

Circle:

First point has x-coordinate 'a':

The y-coordinate must be '1/a':

This is where we see that something is up. When we started this problem, we may have assumed we had lots of choices for 'a'. Alas, there are only four. I won't enumerate them all. Here's one:

Since there are so few of these points, what may have appeared to be an amazing result turns out to be rather a hand-picked result of no particular value that comes immediately to mind. Oh well, it was a wonderful exploration. Plus, I'm relatively confident that there is a much simpler way than this brute-force algebraic solution to produce the desired result, 1.