Two objects, A and B, are moving along two different straight lines at constant speeds. With reference to a particular coordinate system in which distance is measured in metres, the position of A at time t (in minutes) is (4t+2, t-1), and the position of B is (2t + 5,3t-1). (a) Find the equation of the line that A moves along. (b) Let d be the distance between A and B at time t. Show that an expression for d² in terms of t is given by d² = 8t² - 12t +9. (c) A student attempts to find the shortest distance between A and B as shown below. There are two lines where the working does not follow on from the previous line. d² = 8t² - 12t +9 = = 8(t²-t) +9 = 8(t)²- +9 = 8(t - 3)² + 135. The minimum value of d² occurs when t = 1. Hence the minimum distance is 8.4 m (to 2 s.f.). Find and describe the two mistakes. Write out a correct solution, stating when the minimum value of d² occurs and the shortest distance between A and B.

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