Locus Lo Kamar

Algebra Level 5

Consider a point \(\displaystyle P(x,y)\), which moves on the \(\displaystyle xy\) plane on the path \(\displaystyle y=|\alpha x-1|+|\alpha x-2|+\alpha x\), where \(\displaystyle \alpha= \sin \theta\), and \(\displaystyle \theta \in \Big[0,\dfrac{\pi}{2}\Big]\), never letting it's abscissa exceed \(\displaystyle 2\). Let \(\displaystyle A\) be the area of the region \(\displaystyle R\) consisting of all the points \(\displaystyle P\) lying in the first quadrant of the plane when \(\displaystyle \theta\) can take all its values \(\displaystyle \theta \in \Big[0,\dfrac{\pi}{2}\Big]\), then find \(\displaystyle \dfrac{6}{A}\).

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