To find: The exact location of the ship tracked by

Procedure used:
An Application of a Nonlinear System:
"Step 1: Write a system of equations modeling the conditions in the problem.
Step 2: Solve the system and answer the question asked in the problem.
Step 3: Check the proposed solution in the original wording of the problem."
Calculation:
It is given that the ship lies on a path described by ${\displaystyle 2y^2-x^2=1}$ but the boat located on a different path is described by ${\displaystyle 2x^2-y^2=1}$ when the process is repeated.
Use the above procedure to find the exact location of the ship.
Multiply ${\displaystyle 2y^2-x^2=1}$ by 2 and obtain the equation ${\displaystyle 4y^2-2x^2=2}$.
Add the both equations and obtain the result as follows.
${\displaystyle (4y^2-2x^2)+(2x^2-y^2)=2+1}$
${\displaystyle (4y^2-y^2)+(2x^2-2x^2)=3}$
${\displaystyle 3y^2=3}$
${\displaystyle y^2=1}$
${\displaystyle y=\pm 1}$.
Substitute ${\displaystyle y=1}$ in the equation ${\displaystyle 2x^2-y^2=1}$ and obtain the value of x as follows.
${\displaystyle 2x^2-{\left(1\right)}^2=1}$
${\displaystyle 2x^2-1=1}$
${\displaystyle x^2=\frac{1+1}{2}}$
${\displaystyle x^2=1}$
${\displaystyle x=\pm 1}$
Substitute ${\displaystyle y=-1}$ in the equation ${\displaystyle 2x^2-y^2=1}$ and obtain the value of x as follows.
${\displaystyle 2x^2-{(-1)}^2=1}$
${\displaystyle 2x^2-1=1}$
${\displaystyle x^2=\frac{1+1}{2}}$
${\displaystyle x^2=1}$
${\displaystyle x=\pm 1}$.
Thus, for ${\displaystyle y=1}$, $x=\pm 1$ and for ${\displaystyle y=-1,x=\pm 1}$.
Thus, the solution set is ${\displaystyle \{(1,1),(-1,1),(1,-1),(-1,-1)\}}$
However, the ship is located in the first quadrant of the coordinate system.
Hence, the point is (1,1).
Check the result, by substituting the obtained solutions in the given original equations ${\displaystyle 2y^2-x^2=1}$ and ${\displaystyle 2x^2-y^2=1}$.
Substitute (1,1) in the given system and check.
${\displaystyle 2{\left(1\right)}^2-{\left(1\right)}^2=1}$
${\displaystyle 1=1}$
${\displaystyle 2{\left(1\right)}^2-{\left(1\right)}^2=1}$
${\displaystyle 1=1}$
Therefore, the exact location of the ship is (1,1).