# Thread: Word problem using hyperbola...

1. ## Word problem using hyperbola...

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)|

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

C) If A is due north of B, and if P is due east of A, how far is P from A?

2. 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?