Question

as a ship navigates around some rocks in a hyperbolic path, it receives a signal from two transmitters that are 10 kilometers apart. as the ship passes the horizontal line between the beacons, it is 3 kilometers from a buoy which is floating halfway between the transmitters. in this problem, c is the distance from the buoy to transmitter b, and a is the distance from the buoy to where the ship crosses the horizontal line between the transmitter a and transmitter b. which equation represents the path of the ship? a. $\frac{y^2}{9} - \frac{x^2}{16} = 1$ b. $\frac{x^2}{16} - \frac{y^2}{9} = 1$ c. $\frac{y^2}{16} - \frac{x^2}{9} = 1$ d. $\frac{x^2}{9} - \frac{y^2}{16} = 1$

Explanation:

Step1: Define hyperbola parameters

The transmitters are foci, so $2c=10 \implies c=5$. The constant distance difference is $2a=6 \implies a=3$.

Step2: Calculate $b^2$

Use hyperbola relationship $c^2=a^2+b^2$.
$b^2 = c^2 - a^2 = 5^2 - 3^2 = 25 - 9 = 16$

Step3: Write hyperbola equation

Since the hyperbola opens horizontally (ship moves along horizontal axis relative to foci on x-axis), the standard form is $\frac{x^2}{a^2} - \frac{y^2}{b^2}=1$. Substitute $a^2=9$, $b^2=16$:
$\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$

Answer:

B. $\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1$