Problem 48 - College Algebra

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 exact location of the ship is (1,1).

Step 1: Understand the Equations

The two equations given are \(2y^2 - x^2 = 1\) and \(2x^2 - y^2 = 1\). These equations represent two parabolas where the ship might lie. The task is to find an intersection point of these parabolas which would be the exact location of the ship.

Step 2: Solve the Equations

To find the position of the ship, the two equations need to be solved simultaneously. Express \(x^2\) in terms of y from the first equation, and substitute this into the second equation. This gives: \(x^2=2y^2-1\). Substituting into the second equation, you get: \(2(2y^2-1)-y^2=1 => 4y^2-y^2-2=1 => 3y^2-2=1 => 3y^2=2+1 => 3y^2=3 => y^2=1. \) By taking the square root of both sides (Remembering it's in the first quadrant, so y value must be positive) => \(y=1 \).

Step 3: Find the x-coordinate

Now we found y and we can substitute it into one of the original equations to find x-coordinate. Substitute y = 1 into the first equation: \(2(1)^2-x^2=1 => 2-x^2=1 => x^2=2-1 =>x^2=1. \) By taking square root of both sides (Remembering it's in the first quadrant, so x value must be positive) => \(x=1\).