Locus Lo Kamar

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 $\displaystyle \dfrac{6}{A}$ .