I am currently studying multivariate calculus at university, and ive been given some practice problems before the first assignment.
The problem is:
A hyperbola may be defined as the set of points in a plane, the
difference of whose distances from two fixed points $F_1$ and $F_2$ is a
constant. Let P be a point on the hyperbola. Suppose the foci of the
hyperbola are located at (0, ±c), and that $|P F_1| − |P F_2| = ±2a$. It
may be shown that the equation of the hyperbola is given by
$\frac{y^2}{a^2} – \frac{x^2}{b^2} = 1, where \space c^2 = a^2 + b^2$Hyperbolas have many useful applications, one of which is their use in
navigation systems to determine the location of a ship. Two
transmitting stations, with known positions transmit radio signals to
the ship. The difference in the reception times of the signals is used to
compute the difference in distance between the ship and the two transmitting stations. This
infomation places the ship on a hyperbola whose foci are the transmitting stations.
Suppose that radio stations are located at Tanga and Dar es Salaam, two cities on the
north-south coastline of Tanzania. Dar-es Salaam is located 200 km due south of Tanga (you
may assume that Dar es Salaam is directly south of Tanga). Simultaneous radio signals are
transmitted from Tanga and Dar es Salaam to a ship in the Indian Ocean. The ship receives
the signal from Tanga 500 microseconds (µs) before it receives the signal from Dar es Salaam.
Assume that the speed of radio signals is 300m/µs.(a) By setting up an xy-coordinate system with Tanga having
coordinates (0, 100), determine the equation of the hyperbola on which
the ship lies.
(b) Given that the ship is due east of
Tanga, determine the coordinates of the ship.
If someone wouldnt mind giving me a few hints as to how I could solve this, I would be very grateful.
Thanks
Tim
Best Answer
It can be shown that the difference between the two distances (from ship to transmitting stations) is $2a$, where $a$ is the parameter appearing in the hyperbola equation. Let the distance between foci (that is between transmitting stations) be $2c$: you have then the relation $c^2=a^2+b^2$, whence you can compute $b$ and so determine the required equation. Of course you must also set the coordinate system so that Dar es Salaam has coordinates $(0,-100)$.