Word problem using hyperbola...

• Apr 8th 2010, 06:46 PM
uselessjack
Word problem using hyperbola...
Here's a drawing of the provided figure:

http://imgur.com/wrzYH.png

( 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?
• Apr 9th 2010, 12:19 AM
Hello uselessjack
Quote:

Originally Posted by uselessjack
Here's a drawing of the provided figure:

http://imgur.com/wrzYH.png

( 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 $\frac{960\times2640}{5280}=480$ miles.
Quote:

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
$\frac{y^2}{a^2}-\frac{x^2}{b^2}=1$
and the absolute difference between the distances of any point on the hyperbola from the foci is $2a$. So here, $a = 240$.

The foci are at $(0, \pm ae)$. So
$ae = 260$

$\Rightarrow a^2e^2=260^2$
But
$a^2e^2 = a^2+b^2$
$\Rightarrow b^2= a^2e^2 - a^2$
$=260^2-240^2$
$\Rightarrow b = 100$
Can you finish off now?