Question

LORAN is a long range hyperbolic navigation system. Suppose two LORAN transmitters are located at the coordinates (−150,0) and (150,0), 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 200 miles.

What is the standard form of the hyperbola that the receiver sits on if the transmitters behave as foci of the hyperbola?

Answer 1

x²/10000 -y²/12500 = 1

Explanation:

You want the standard form equation of the hyperbola that is the locus of points with a difference of 200 miles from foci at (-150, 0) and (150, 0), where each grid unit is one mile.

Hyperbola

The standard form equation of a hyperbola with semi-transverse axis 'a' and semi-conjugate axis 'b' is ...

x²/a² -y²/b² = 1

The relationship between these semi-axes and the distance from the center to the focus (c) is ...

a² +b² = c²

The relationship between the difference of distances to the foci (d) and these other parameters is ...

a = d/2

Application

The foci are located on the x-axis at (±c, 0).

For the given values c = 150 and d = 200, we can find 'a' and 'b' as ...

a = d/2 = 200/2 = 100

b² = c² -a² = 150² -100² = 12500

The equation of the desired hyperbola is ...

x²/10000 -y²/12500 = 1