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Math Help - Word problem using hyperbola...

  1. #1
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    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?
    Last edited by uselessjack; April 8th 2010 at 06:05 PM.
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  2. #2
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    Grandad's Avatar
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    Hello uselessjack
    Quote Originally Posted by uselessjack View Post
    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 \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
    \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?

    Grandad
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