Let the length of a latus rectum of an ellipse \(\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\) be \(10\). If its e…

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