The first chord passing through the point P will be from (29,0) to (-29,0) with a length of 58.
Another chord will be from (-29,0) to (29,0) or in the opposite direction, but will still have the same length of 58.
I suppose that only makes a chord count (with integer length ) of 1 since they both are equal to 58.
That is the longest possible chord.
The shortest possible chord will occur when a point on the circle occurs at (20,y) or (20,-y).
[That creates a right triangle]
The radius (or hypotenuse) is 29,
the base is 20,
the height is y
Thus the chord will be from (20,21) to (20,-21) for an integer length of 42.
That is the shortest chord possible.
The chord length will vary (continously) from 42 to 58. You will have a chords of various lengths between 42 to 58 inclusive.
Simply count the number of integers from 42 to 58 (including 42 and 58).
That should be your answer
Here's a picture to illustrate the answer. The longest chord (length 58) is the diameter AB. The shortest is the perpendicular chord CD (length 42). These only occur once. Every intermediate length occurs twice, once as X moves along the arc from B to C, and once as X moves along the arc from C to A. The total number is that given by Isomorphism.