The use of the LORAN navigation system is currently diminished by the use of the GPS

Chapter 14, Problem 14.5.43

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The use of the LORAN navigation system is currently diminished by the use of the GPS system, but LORAN continues being used as a backup system to GPS. The technique for finding your position is to measure the time for a wave to be sent and returned from three transmitters. Using a pair of transmitters as foci it is possible to find a hyperbolic curve on which your position coordinates lie. Using a second pair of foci, a second curve is found and your position is the intersection of the two hyperbolic curves. We use the property that the difference between the distance from two foci is constant for a hyperbola. However, for convenience in this exercise, we replace time measure with a distance measure. Suppose that you are at position P in Figure 14.50 and that the three transmitters lie along the vertical axis as shown. (a) Find the equation of a hyperbola that passes through P and has foci T1 and T2. (b) Find the equation of a hyperbola that passes through P and has foci T1 and T3. (c) Solve for y in the equations you found in parts (a) and (b). In each case choose the branch that passes through P. (d) Use a calculator or computer to graph the branches of the hyperbolas from part (c).