$16 \cdot 22$ Let point $P$ be at a distance of 9 from the center of a circle with radius 15. The number of chords passing through $P$ and having integer lengths in the circle is
(A) 11.
(B) 12.
(C) 13.
(D) 14.
(E) 29.
(42nd American High School Mathematics Examination, 1991)
[Solution] From the given, the shortest chord of the circle passing through point $P$ is $\dot{2} \sqrt{15^{2}-9^{2}}=24$, and the longest one (i.e., the diameter of the circle passing through point $P$) is 30.
There is only one each of the longest and shortest chords.
Chords of lengths $25, 26, 27, 28, 29$ each have two.
Thus, there are 12 chords passing through $P$ with integer lengths.
Therefore, the answer is $(B)$.