Dancing on a parabola Two people, A and B, walk along the

Chapter 4, Problem 78

(choose chapter or problem)

Dancing on a parabola Two people, A and B, walk along the parabola y = x2 in such a way that the line segment L between them is always perpendicular to the line tangent to the parabola at As position. What are the positions of A and B when L has minimum length? a. Assume that As position is 1a, a22, where a 7 0. Find the slope of the line tangent to the parabola at A and find the slope of the line that is perpendicular to the tangent line at A. b. Find the equation of the line joining A and B when A is at 1a, a22. c. Find the position of B on the parabola when A is at 1a, a22. d. Write the function F 1a2 that gives the square of the distance between A and B as it varies with a. (The square of the distance is minimized at the same point that the distance is minimized; it is easier to work with the square of the distance.) e. Find the critical point of F on the interval a 7 0. f. Evaluate F at the critical point and verify that it corresponds to an absolute minimum. What are the positions of A and B that minimize the length of L? What is the minimum length? g. Graph the function F to check your work. B A L O y x y _ x2 Additional Exercises