Question

(score for question 2: ___ of 5 points)

1. 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:

Explanation:

Step1: Identify hyperbola parameters

Foci at $(\pm100,0)$: $c=100$. Difference of distances: $2a=180 \Rightarrow a=90$.

Step2: Calculate $b^2$

Use $c^2=a^2+b^2$: $b^2=100^2-90^2=1900$.

Step3: Write standard form

Horizontal hyperbola: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$.

Answer:

$\frac{x^2}{8100} - \frac{y^2}{1900} = 1$