Hello uselessjack Originally Posted by

**uselessjack** Here's a drawing of the provided figure:

( URL of the picture in case it doesn't display:

http://imgur.com/wrzYH.png )

The axes x and y are measured in miles.

In the figure, the LORAN stations at A and B are 520 mi apart, and the ship at P receives station A's signal 2,640 microseconds (ms) before it receives the signal from B.

A) Assuming that radio signals travel at 960 ft/ms, find | d(P, A) - d(P, B)|

The difference between the distances AP and PB is $\displaystyle \frac{960\times2640}{5280}=480$ miles.

B) Find an equation for the branch of the hyperbola indicated in red in the figure, using miles as the unit of distance.

The equation of a North-South opening hyperbola, centre the origin, is of the form

$\displaystyle \frac{y^2}{a^2}-\frac{x^2}{b^2}=1$

and the absolute difference between the distances of any point on the hyperbola from the foci is $\displaystyle 2a$. So here, $\displaystyle a = 240$.

The foci are at $\displaystyle (0, \pm ae)$. So

$\displaystyle ae = 260$

$\displaystyle \Rightarrow a^2e^2=260^2$

But

$\displaystyle a^2e^2 = a^2+b^2$

$\displaystyle \Rightarrow b^2= a^2e^2 - a^2$$\displaystyle =260^2-240^2$

$\displaystyle \Rightarrow b = 100$ Can you finish off now?

Grandad