Coordinate Geometry: JEE Main 2025 Mathematics question

The centre of a circle $C$ is at the centre of the ellipse $E:\;\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ such that the circle $C$ and the ellipse $E$ intersect at four points. Let $P$ be one of these four points. If the area of the triangle $\triangle PF_1F_2$ is $30$ and the length of the major axis of $E$ is $17$, then the distance between the foci of $E$ is: