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# LORAN question

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LORAN is a long range hyperbolic navigation system. Suppose two LORAN transmitters are located at the coordinates (-100,0) and (100,0) where unit distance on the coordinate plane is measured in miles A receiver is located somewhere in the first quadrant. The receiver computes that the difference in the distances from the receiver to these transmitters is 180 miles.

What is the standard form of the hyperbola that the receiver sits on if the transmitters behave as foci of the hyperbola? Show all steps of your work as evidence of the solution.

Nov 12, 2019

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The center of the hyperbola  is  (0,0)  = (h, k)

c = the dist ance form the center to either focal point  = 100

c^2 =  100^2   =  10000

The differences  from the receiver to the transmitters  =  2a

So

2a =  180

a   =  90

a^2 = 8100

And

b^2  = c^2 - a^2

b^2  = 10000 - 8100

b^2  =1900

The form is

(x - h)^2              (y - k)^2

_______   +          ______    =      1

a^2                          b^2

(x - 0)^2               (y - 0)^2

_______   +        _______  =    1

10000                  1900

x^2                   y^2

______       +    _____   =    1

10000               1900

Nov 13, 2019