Question
An airplane leaves Doha airport bound for Paris Charles de Gaulle airport. There are lights located at the end of the runway at points \(A(0,4)\) and \(B(6,8)\), relative to a terminal at the origin. The takeoff path of the airplane is the perpendicular bisector of line \(A B\).
(a) Find the equation of the takeoff path in the form \(a x+b y+d=0\), where \(a, b, d \in \mathbb{Z}\).
The airplane travels at an average speed of \(570 \mathrm{~km} \mathrm{hr}^{-1}\) in a straight line. Once the airplane has reached cruising altitutde, an air traffic controller at the top of a \(200 \mathrm{~m}\) high air traffic control tower at \(C(7,0)\) observes that the angle of elevation to the airplane is \(40^{\circ}\). Five minutes later, the controller observes that the angle of elevation is \(10^{\circ}\).
(b) Find the cruising altitude of the airplane in metres.
As the airplane is about to land at the Paris Charles de Gaulle airport, the pilot is asked to delay the landing due to a traffic issue. The pilot is instructed to turn the airplane on a bearing of \(045^{\circ}\) for \(10 \mathrm{~km}\) until reaching point \(\mathrm{P}\), then travel on a bearing of \(165^{+}\)for \(30 \mathrm{~km}\) to point \(\mathrm{Q}\) before flying back to the original point \(\mathrm{O}\) for landing.
(c) Find the angle \(O \hat{P Q}\).
(d) Find the shortest distance from \(\mathrm{Q}\) back to \(\mathrm{O}\) for landing.
(e) Find the bearing the airplane must travel on to get back to \(\mathrm{O}\) from \(\mathrm{Q}\).