The equation of the hyperbola on which the ship is positioned is x²/2025-y²/4375=1
Given that radio stations, A and B are 160 kilometers apart and the signal from B arrives 0.0003 seconds before the signal from A and the signal travels 300,000 kilometers per second.
The distance between radio station A and B are d=160 km.
The time is t=0.0003s.
The speed of the signal travel is v=300000 km/s.
Consider both radio stations A and B lie at foci on the x-axis.
The origin lies at the midpoint between the stations.
The distance of one of the stations from the origin is,
c=d/2
c=160km/2
c=80km
The centre of the hyperbola lies at (0,0), so both stations lie at coordinates points of (-80,0) and (80,0).
The expression for the difference in distance is,
I=vt
I=300000×0.0003
I=90km
The semi-major axis of the hyperbola is,
a=I/2
a=90km/2
a=45km
The expression for the semi-minor axis of the hyperbola is,
b²=c²-a²
b²=(80)²-(45)²
b²=6400-2025
b²=4375
The expression for the hyperbola on which the ship is positioned,
x²/a²-y²/b²=1
x²/(45)²-y²/4375=1
x²/2025-y²/4375=1
Hence, the equation of hyperbola on which the ship is positioned for radio stations, A and B are 160 kilometers apart and signal from B arrives 0.0003 seconds before the signal from A is x²/2025-y²/4375=1.
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