Let C be the circle of minimum area enclosing the ellipse E: x^2/a^2+y^2/b^2=1 with eccentricity 1/2 and foci ( ± 2,0). Let P Q R be a variable…

Let $\displaystyle C$ be the circle of minimum area enclosing the ellipse $\displaystyle E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\displaystyle \frac{1}{2}$ and foci $\displaystyle ( \pm 2,0)$. Let $\displaystyle P Q R$ be a variable triangle, whose vertex $\displaystyle P$ is on the circle $\displaystyle C$ and the side $\displaystyle Q R$ of length $\displaystyle 2 a$ is parallel to the major axis of $\displaystyle E$ and contains the point of intersection of $\displaystyle E$ with the negative $\displaystyle y$-axis. Then the maximum area of the triangle $\displaystyle P Q R$ is :