To find the exact location of the ship in the first quadrant, we need to solve the following system of equations:
[tex]\[
4y^2 - x^2 = 12 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
3x^2 - y^2 = 8 \quad \text{(Equation 2)}
\][/tex]
Both [tex]\( x \)[/tex] and [tex]\( y \)[/tex] must be positive since the ship is located in the first quadrant.
1. Let's start with Equation 2 and express [tex]\( y^2 \)[/tex] in terms of [tex]\( x \)[/tex]:
[tex]\[
y^2 = 3x^2 - 8
\][/tex]
2. Substitute [tex]\( y^2 \)[/tex] from this expression into Equation 1:
[tex]\[
4(3x^2 - 8) - x^2 = 12
\][/tex]
3. Expand and simplify the equation:
[tex]\[
12x^2 - 32 - x^2 = 12
\][/tex]
[tex]\[
11x^2 - 32 = 12
\][/tex]
[tex]\[
11x^2 = 44
\][/tex]
[tex]\[
x^2 = 4
\][/tex]
[tex]\[
x = 2
\][/tex]
Since we are only considering the first quadrant, we take the positive root, [tex]\( x = 2 \)[/tex].
4. Now, substitute [tex]\( x = 2 \)[/tex] back into the expression for [tex]\( y^2 \)[/tex]:
[tex]\[
y^2 = 3(2)^2 - 8
\][/tex]
[tex]\[
y^2 = 3 \cdot 4 - 8
\][/tex]
[tex]\[
y^2 = 12 - 8
\][/tex]
[tex]\[
y^2 = 4
\][/tex]
[tex]\[
y = 2
\][/tex]
Again, since we are in the first quadrant, we take the positive root, [tex]\( y = 2 \)[/tex].
Thus, the exact location of the ship is [tex]\((2, 2)\)[/tex].
[tex]\[
\boxed{(2, 2)}
\][/tex]
