LORAN follows an hyperbolic path.
The equation of the hyperbola is:
The coordinates are given as:
The center of the hyperbola is
The distance from the center to the focal points is given as:
Square both sides
The distance from the receiver to the transmitters is given as:
Divide both sides by 2
Square both sides
We have:
This gives
The equation of an hyperbola is:
So, we have:
Hence, the equation of the hyperbola is:
Read more about hyperbolas at:
We see that in these 3 terms, 6 is a common factor. So, let's factor out a 6:
We can set this equal to 0 and factor by using the quadratic formula which is:
So, let's do just that:
Note that the 6 goes away if you divide both sides by 6. In this case, a = 1, b = -1, and c = -2. Let's plug that into the quadratic equation:
So, we can write this as:
Notice that the 6 comes back because it was only temporarily mad. And that the roots have opposite signs in the parentheses because to find the roots, you need to set each parentheses equal to 0 and solve for n. With that in mind, your final answer is 6(n-1)(n+2). Hope I could help you!