Let the length of a latus rectum of an ellipse \( \dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 \) be \( 10 \). If its eccentricity is the minimum value of the function \( f(t) = t^{2} + t + \dfrac{11}{12},\, t \in \mathbb{R} \), then \( a^{2} + b^{2} \) is equal to:
