Airplanes are equipped with signal devices to alert rescuers to their position. Suppose a downed plane sends out radio pulses that are detected by two receiving stations, A and B. The stations are located 130 miles apart along a stretch of I-40, with A due west of B. The two stations are able to determine that the plane is 50 miles farther from station B than from station A.
a. Determine the equation of the hyperbola centered at the origin on which the plane is located.
b. Graph the equation, indicating on which branch of the hyperbola the plane is located.
c. If the pilot estimates that the plane is 6 miles from I-40, find the exact coordinates of its position.