In this question, positions are given with reference to a Cartesian coordinate system whose x- and y-axes point due East and due North, respectively. Distances are measured in kilometers.
A light aircraft travels at a constant speed of 480 kilometers per hour (kph) in a straight line from airport A, located at position (xA, yA) = (−200, 100), to airport B at (xB, yB) = (400,−100).
(i) Find the equation of the line of travel of the aircraft.
(ii) Find the direction of travel of the aircraft, as a bearing, with the angle in degrees correct to one decimal place.
(iii) Calculate the distance between airports A and B, in kilometers, correct to the nearest kilometer.
(iv) How long does the aircraft take to travel from A to B? Give your answer in hours and minutes, correct to the nearest minute.
(v) Find parametric equations for the line of travel of the aircraft. Your equations should be in terms of a parameter t, and should be such that the aircraft is at airport A when t = 0 and at airport B when t = 1.
(b) During its journey, the aircraft passes a landmark, L, located at position (90,−30).
(i) Let d kilometers be the distance between the location of the aircraft at parameter value t and the landmark L. Find an expression for d2 in terms of t. Simplify your result as far as possible.
(ii) Using your answer to part (b)(i) and the method of completing the square, determine the shortest distance, to the nearest kilometer, between the aircraft and the landmark L.
(iii) Landmark L is visible from a distance of at most 100 kilometers. Calculate the parameter values t1 and t2 when the landmark L can first and last be seen from the aircraft. For how many minutes is the landmark visible from the aircraft?
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