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 A’s position. What are the positions of A and B when L has minimum length?

a. Assume that A’s position is (a, a²), where a > 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 (a, a²).

c. Find the position of B on the parabola when A is at (a, a²).

d. Write the function F (a) 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 > 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.

Transcribed Image Text:

y У — 2 B х