Question

LORAN is a long-range hyperbolic navigation system. Suppose two LORAN transmitters are located at the coordinates and, 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.

Answer 1

x²/10000 + y²/1900 = 1

Step-by-step explanation:

coordinates (-100,0) and (100,0)  missing in Question

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

c = the distance form the center to either focal point  = 100

c² =  100²   =  10000

The differences  from the receiver to the transmitters  =  2a

2a =  180

=> a   =  90

=> a² = 8100

b²  = c² - a²

b²  = 10000 - 8100

b²  =1900

(x - h)²/a²  + (y - k)²/b²  = 1

=> x²/10000 + y²/1900 = 1

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Answer 2

Answer:

Step-by-step explanation:

Missing part in question :-

transmitters are located at the coordinates (-100,0) and (100,0)

Answer :-

The hyperbole will horizontal based.

For horizontal hyperbola :-

(x - h)² / a² - (y - k)² / b² = 1

Since the foci are (-100,0) and (100,0) & difference in the distances from the receiver to these transmitters is 180 miles, so

Center here (h,k) will be (0,0)

Based on above 2 points,

vertices will be (-90,0) and (90,0)

There fore, a = 90,

b = √(100² - 90²) = √1900

So, standard form of hyperbole will be :-

x ² / 8100   -    y² / 1900 = 1