In the 1600s, René Descartes married algebra and geometry to create the Cartesian plane.

Point \(A\) and \(B\) always has a constant distance of \(1\) from each other.

Point \(A\) moves with a horizontal velocity (parallel to the x-axis) of \(v\)

Point \(B\) moves with a horizontal velocity of \(2v\)

Point \(B\)'s vertical velocity (parallel to the y-axis) is \(0\) while Point \(A\) is allowed to move vertically in order to keep the constant distance of \(1\).

All this movement is happening in the first quadrant.

Given that the area made with the \(x\) and \(y\) axis and the path traveled by Point \(A\) is \(P\), find \[\left\lfloor 1000P \right\rfloor \]

After you solve this, you might want to try a continuation of this problem.

Try my Other Problems

If \(AB=36\) and \(BC=50\), then the maximum possible value of \(BQ\) can be written in the form \(a-\sqrt{b}\), where \(a,b \in \mathbb{N}\). What is \(a+b\)?

\[ (ax-by)^2 + (bx-ay)^2 = c^2 \]

For positive integers \(a,b,c\). What is the minimum value of \(a+b+c\)

**Details and Assumptions**

- Diagram not up to scale

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