Two LORAN radio stations A and B are 110 kilometers apart and send a simultaneous radio signal to a ship.

Question

Two LORAN radio stations A and B are 110 kilometers apart and send a simultaneous radio signal to a ship.

The signal from B arrives 0.0001 seconds before the signal from A. If the signal travels 300,000 kilometers per second, find the equation of the hyperbola on which the ship is positioned. Use a coordinate system where the origin is the midpoint between the stations, and the stations lie on the x-axis.

Answer 1

v=300,000km/sec

a=110

b=299,999.99.........km/sec

c=300,000km/sec

You want α.

α=acos((b^2+c^2-a^2)/(2bc)

α=.210846..........°

Answer 2

Difference of distances 2a = 300000km/s·0.0001s= 30 km; SA - SB = 30km = 2a = 30 km; a = 15 km

Points A and B are foci of hyperbola, S is ship; so 2c = 110 km; c = 55km

Then b² = c² - a² = 55² - 15² = 2800.

Now equation of hyperbola x²/a² - y²/b² = 1 or x²/225 - y²/2800 = 1