Coordinate Geometry: JEE Main 2025 Mathematics question

Let $C$ be the circle of minimum area enclosing the ellipse $\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $\tfrac12$ and foci $(\pm2,0)$. Let $PQR$ be a variable triangle whose vertex $P$ lies on $C$ and whose side $QR$ of length $29$ is parallel to the major axis of the ellipse and passes through the point where the ellipse intersects the negative $y$–axis. Then the maximum area of $\triangle PQR$ is: