Answer Happy

(13 pts.) Given a circle of radius a centered at Olet AB be a diameter and P an arbitrary point on the circle. Draw another circle of radius r = AP centered at A and let O be the point where this circle meets the segment AB. The main object of this problem is to determine where P should be located so as to maximize the area of the triangle APQ. (a) Let a denote the measure of the angle PAQ. Show that no matter where P is located, a equals the angle of intersection between the (tangents to the) two circles at P. b) Find a formula for the area of triangle APQ in terms of r anda , then eliminate a to get the area as a function of r alone. (Your formula will of course also contain the constant a.) 2) Apply differential calculus to the function you found in part (b) to determine the value ofr that maximizes the area of triangle APQ. What is the area in this case? How about a ?