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?
Answer:

[tex]\mathrm{The \ equation \ of \ the \ hyperbola\ is}\displaystyle \ \underline{\frac{y^2}{90^2} - \frac{x^2}{10\left(\sqrt{19} \right)^2} = 1}[/tex]

Reasons:

The location of the transmitters = (-100, 0) and (100, 0)

The difference in the distance from the receiver to the transmitters = 180 miles.

Let the distances from the receiver to the transmitters be d₁ and d₂, we have;

|d₂ - d₁| = 180 = 2·a

[tex]\displaystyle a = \frac{180}{2} = 90[/tex]

c = The x-coordinates of the transmitters = 100

b² = c² - a²

∴ b² = 100² - 90² = 1900

b = √(1900) = 10·√(19)

Therefore;

The general form of the equation of an hyperbola is presented as follows;

[tex]\displaystyle \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1[/tex]

The standard form of the hyperbola that the receiver sits on if the transmitters behave as foci of the hyperbola is; [tex]\displaystyle \underline{\frac{y^2}{90^2} - \frac{x^2}{10\left(\sqrt{19} \right)^2} = 1}[/tex]

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