In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points] (1) $\sqrt{43}$ (2) $\sqrt{47}$ (3) $\sqrt{51}$ (4) $\sqrt{55}$ (5) $\sqrt{59}$
In coordinate space, there is a right triangle ABC with $\overline{\mathrm{AB}} = 8$, $\overline{\mathrm{BC}} = 6$, $\angle\mathrm{ABC} = \frac{\pi}{2}$ and a sphere $S$ with diameter AC. Let $O$ be the circle formed by the intersection of sphere $S$ with the plane that contains line AB and is perpendicular to plane ABC. Let P and Q be two distinct points on circle $O$ such that the distance from each to line AC is 4. Find the length of segment PQ. [4 points]\\
(1) $\sqrt{43}$\\
(2) $\sqrt{47}$\\
(3) $\sqrt{51}$\\
(4) $\sqrt{55}$\\
(5) $\sqrt{59}$