You can, without loss of generality, assume that the center of the circle is at (0, 0) on a coordinate system and that P is at (0, 6). Now the circle can be written as and any line through P as y= mx+ 6. Put that into the quadratic and solve for x to determine the points where the line crosses the circle (depending on m). Find the length of the line segment between the points (still depending on m). What values of m give an integer length?