You can draw a line of any desired gradient (including m = 1) through any given point (including points in S).
So I would've thought the answer would be equal to the number of points in S .....
Do you mean how many tangents to either xy = 2 or ....? Or, perhaps how many lines passing through two of the points in S ....?
Anyway, you certainly need to know how many points are in S ......
Let , say:
Therefore and so there are two solutions for x. Therefore there are two points in S.