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Rose-Hulman Math/Physics Challenge

Given a point P and a line segment AB, determine the point Q on AB which a bead sliding under the influence of gravity (without friction or any other resistance) on the segment PQ will reach first.
Doofenshmirtz 400  
To put this into a real-world context, imagine that Agent P is traveling by zip-line to the AB building to meet Agent Q. Agent P wants to get from his location at the top of building X to the AB building as fast as possible, so Agent P needs to locate the end of his zip-line (shoot an arrow) at what point of the AB building and tell Agent Q to meet him there. As a first case, we consider a frictionless zip-line and no resistance to Agent P's motion on the zip-line, with the motion being purely under the influence of gravity (and no deflection of the line by Agent P's weight).

HINTS: For convenience, set point P = (0,p), point A = (a,p) and point B = (b,0). Treat numbers a, b, and p as positive numbers.
Bonus Problem 1:
How does the result change with a frictional force on the zip-line with the frictional force opposing the motion and proportional to the component of the weight that is perpendicular to the line PQ?
Bonus Problem 2:
How does the result change with a resistance force proportional to the velocity of Agent P's motion?

Research Question:
How would one account for the deflection of the line under the weight of Agent P? What information would you need to know about the zip-line?

REMARKS: The original problem and bonus problem 1 only require geometry, algebra and Newton's laws of motion to solve. Bonus problem 2 requires calculus to set up and solve. The research question requires even more sophisticated mathematics and physics to set-up and solve.