Acircle is drawn in the xy-coordinate plane. if there are n different points (x, y) on the circle such that xy = 0, then the possible values of n are

For xy to be 0, where x and y are the first and second coordinates of a point in the coordinate axes, 

either x, or y or both must be 0.

So we consider the points of the form (x, 0), (0, y) and (0, 0). These are the x and y intercepts.

A circle can touch the axes at 1, 2, 3 or 4 points but not more.

Check the picture for examples of all the cases.

n∈{1, 2, 3, 4}