Coordinate Geometry: JEE Main 2026 Mathematics question

Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b)$, be 30. If its eccentricity is the maximum value of the function $f(t)=-\frac34+2t-t^2$, then $(a^2+b^2)$ is equal to