Analytic Geometry help!?

Question

A and B are two loran stations located at (-500,0) and (500,0), respectively. A ship's receiver detects radio signals sent simultaneously from the two stations and indicates that the ship is 600 miles closer to A than to B. Similarly, it is found that the ship is 100 miles closer to C, at (0, 1300), than to D, at (0,-1300). Where is the ship (point)?

Answer 1

The ship is approximately at (-300.65, 26.47).
The set H of points with the property that the difference of distances between each point of H and 2 given points is a hyperbola with these 2 points as focuses. Hence the required point lies on the left branch of the hyperbola:
H1: x²/300² - y²/400² = 1 /focuses A and B, real axis length 600/
and on the upper branch of the hyperbola
H2: -x²/(1300² - 50²) + y²/50² = 1 /focuses C and D, real axis length 100/.
Solving both equations easily follows that H1 and H2 intersect in 4 points (1 per quadrant):
x ≈ ± 300.65, y ≈ ± 26.47
The required point is in the 2nd quadrant /closer to A and C/.

Meanwhile are You sure about those 100 miles? Aren't they 1000? The first 600 are OK, half of it being 300 because
500² - 300² = 400²
and a half of 1000 would produce a perfect square
1300² - 500² = 1200² instead of 1300² - 50² in the H2's equation, the latter would be:
H2: -x²/1200² + y²/500² = 1
/2 very well-known Pythagorean triangles involved/
Of course the answer would be different then:
x = -300*√(656/231) ≈ -505.55
y = 2000*√(17/231) ≈ 542.56