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In a plane $$AB$$ is a fixed line segment of length 8. $$P$$ is a varying point in the same plane such that $$\overline{AB}$$ subtends a right angle at $$P$$ and the length of the perpendicular from $$P$$ on to $$\overline{AB}$$ is 3. Find the number of such points $$P$$.
Correct Answer: 4
$$\angle APB = 90^\circ$$ means $$P$$ lies on the circle with diameter $$AB$$, of radius 4 (excluding $$A$$ and $$B$$). The points at distance 3 from line $$AB$$ lie on two lines parallel to $$AB$$, one on each side. Since $$3 < 4$$, each line cuts the circle in 2 points, giving 4 points in all.
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