Chapter 10.2, Problem 84PE

a. Radio signals emitted from points $\left(8,0\right)$ and $\left(-8,0\right)$ indicate that a plane is $8\text{ mi}$ closer to $\left(8,-0\right)$ than to $(-8,0)$ . Find an equation of the hyperbola that passes through the plane's location and with foci $\left(8,0\right)$ and $\left(-8,0\right)$ . All units are in miles.

b. At the same time, radio signals emitted from points $\left(0,8\right)$ and $\left(0,-8\right)$ indicate that the plane is $4\text{ mi}$ farther from $\left(0,8\right)$ than from $\left(0,-8\right)$ . Find an equation of the hyperbola that passes through the plane's location and with foci $\left(0,8\right)$ and $\left(0,-8\right)$ .

c. From the figure, the plane is located in the fourth quadrant of the coordinate system. Solve the system of equations defining the two hyperbolas for the point of intersection in the fourth quadrant. This is the location of the plane. Then round the coordinates to the nearest tenth of a mile.