Ellipse

Let \(A(\alpha, 0)\) and \(B(0, \beta)\) be the points on the line \(5 x+7 y=50\). Let the point \(P\) divide the line segment \(A B\) internally in the ratio \(7:3\). Let \(3 x-25=0\) be a directrix of the ellipse \(E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) and the corresponding focus be \(S\). If from \(S\), the perpendicular on the \(x\)-axis passes through \(P\), then the length of the latus rectum of \(E\) is equal to,