This line is crossed by four lines. how many points of intersection of these lines with a given one can form?

All four lines can intersect a given line at one point, if their intersection point is on this line – in this case, one intersection point.
Of the four straight lines, two or three straight lines can have an intersection point on a given straight line, in this case there can be two or three intersection points.
If the points of intersection of four lines do not belong to this line, then in this case there will be four points of intersection.
Four lines can form from one to four points of intersection with a given line.