Hyperbola...?

Question

LORAN, long distance radio navigation for aircraft and ships, uses synchronized pulses transmitted by widely separated transmitting stations. These pulses travel at the speed of light (186,000 miles per second). The difference in the times of arrival of these pulses at an aircraft or ship is constant on a hyperbola which has transmitting stations located at the foci. Assume that the two stations, 300 miles apart, are positioned on a rectangular coordinate system at points (-150,0) and (150,0) and that a ship is traveling on a path with coordinates (x, 75).

a. Find the x-coordinate of the position of the ship if the time difference between the pulses from the transmitting station is .0001 second.

b. Write the equation of the hyperbola on which the ship is located

Answer 1

Focus 1 (F1) is at (-150),0 and F2 is at (150,0)
The ship is at a point P(x,75)
The distance F1P is [(x+150)^2 +75^2)]^0.5
The distance F2P is [(x- 150)^2 +75^2)]^0.5

The difference between these two distances is 186,000*.0001=
18.6 miles. Therefore a = 9.3 and a^2=86.49

c=1/2 F1F2=150
b^2= c^2-a^2=22500-86.49=22413.51

The equation of the hyberbola is x^2/86.49-y^2/22,413.51=1
To find x we use the equation:
[(x+150)^2 +75^2)]^0.5 - [(x- 150)^2 +75^2)]^0.5 = 18.6
[x^2+300x+ 28125]^0.5-[x^2 -300x +28125]^.5=18.6
x^2+300x+ 28125= 18.6^2+37.2[x^2 -300x +28125]^.5 +x^2 -300x +28125
600x-345.96 = 37.2[x^2 -300x +28125]^.5
360000x^2 -691.92x +119,688.32 = 1383.84x^2 -41515.2x+38,920,500
358,616.16x^2+40,823.28x -38,800,812
x^2 + .1138x - 108.196=0
x= [-.1138 + sqrt(.1138^2 +432.784)]/2 = 10.35