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 o
f 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
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 $(0,-100)$.