# A submarine on the surface of the ocean, is on a course parallel to and 60 Km from the shore line as shown in the figure. There are two transmitters, X and Y located 200 Km from each other on the shore line. The submarine uses the signals to figure out that its 50 Km closer to station Y than to X. Find the distance between the submarine to each of the stations.

Hint: If $d_x$ and $d_y$ are the distances from the sub to X and Y and $d_x - d_y = 50$, then the ship must be on a hyperbola with foci at X and Y.

$\begin{array}{1 1} (A) d_x = 134, \; d_y = 84\\ (B) d_x = 142.6, \; d_y = 92.6 \\(C) d_x = 182.6, \; d_y = 132.6 \\ (D) d_x = 132.4, \; d_y = 82.4 \end{array}$

Given: Hint: If $d_x$ and $d_y$ are the distances from the sub to X and Y and $d_x - d_y = 50$, then the ship must be on a hyperbola with foci at X and Y, as shown below:
The equation for the hyperbola is $\large\frac{x^2}{a^2} $$- \large\frac{y^2}{b^2}$$ = 1$
Here $a = \large\frac{1}{2}$$50 = 25, the distance from the center to each vertex and c = 100, distance from center to foci X or Y. . \Rightarrow b = \sqrt (c^2 - a^2) = \sqrt (100^2 - 25^2) = \sqrt 9375 \Rightarrow The equation for the hyperbola is \large\frac{x^2}{625}$$ - \large\frac{y^2}{9375}$$= 1 Substitute, y = 60, \rightarrow \large\frac{x^2}{625}$$ - \large\frac{3600}{9375}$$= 1 \Rightarrow x^2 = \large\frac{9375+3600}{9375}$$\times 625 = 865$
$\Rightarrow x = \sqrt 865 \approx 29.4$
Therefore, we can now calculate $d_x$ and $d_y$ as follows:
$d_x = \sqrt (29.41+100)^2 \approx 142.6$ Km
$d_y = \sqrt (29.41 -100)^2 \approx 92.6$ Km
edited Mar 26, 2014