Since $A$ is the midpoint of the chord, the diameter bisects it. The diameter of the circle is $30$ (Using Pythagoras Theorem). It is the shortest chord that passes through $A$ and the longest chord is the diameter. All the integers between $24$ and the diameter i.e. $30$ account for $2$ distinct chords. This is a consequence of Intermediate Value Theorem i.e., the length of the chord is a decreasing function of the smaller of the angles it makes with the diameter. Therefore we have the number of distinct chords as : $1 + 1 + 2\times (30 - 24 - 1) = 12.$
Correct Option: D.