Problem 56 - Algebra and Trigonometry

Question

A system for tracking ships indicates that a ship lies on a path described by \(2 y^{2}-x^{2}=1 .\) The process is repeated and the ship is found to lie on a path described by \(2 x^{2}-y^{2}=1 .\) If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.

Step-by-Step Solution

Answer

The ship's exact location in the first quadrant of the coordinate system is at the point (1, 1)

Step 1: Set the Equations Equal to Each Other

Since the ship lies on both paths, it means that the two equations are equal at the coordinate of the ship. Therefore, set the two equations equal to each other: \(2 y^{2} - x^{2} = 2 x^{2} - y^{2}\)

Step 2: Rearrange the Equation

Rearrange the equation from step 1, with the goal of getting the squares of x terms together and the squares of y terms together. This leads to the equation: \(3 y^{2} = 3 x^{2}\). From this equation, one can deduce that \(y^{2} = x^{2}\) or \(y = x\) or \(y = -x\). However, since we are in the first quadrant wherein both x and y are positive, only the first solution \(y=x\) is possible.

Step 3: Substitute y into original equations

Substitute \(y=x\) into one of the original equations to solve for exactly one of the variables. For instance, substituting \(y=x\) into the first equation gives \(2 x^{2} - x^{2} = 1\), which simplifies to \(x^{2} = 1\), giving \(x = 1\). You can then substitute \(x = 1\) into the equation \(y = x\) to get \(y = 1\).

Step 4: Check the solution

Substitute the obtained values for x and y into both equations to check that they satisfy the equations. Thus, check that \(2*(1)^{2} - (1)^{2} = 1\) and \(2*(1)^{2} - (1)^{2} = 1\). Both these equations are indeed true, therefore the solution obtained is correct.