Let x ^ 2 a ^ 2 + y ^ 2 b ^ 2 = 1 (a > b) be a given ellipse,...

$LR = \dfrac{2{b}^{2}}{a} = 10$ $⇒ {b}^{2} = 5a$

$ϕ\left(t\right) = \dfrac{5}{12} - \left({t}^{2} - t + \dfrac{1}{4} - \dfrac{1}{4}\right) = \dfrac{5}{12} + \dfrac{1}{4} - {\left(t - \dfrac{1}{2}\right)}^{2}$

$= \dfrac{2}{3} - {\left(t - \dfrac{1}{2}\right)}^{2}$

max $ϕ\left(t\right) = \dfrac{2}{3} = e$

${b}^{2} = {a}^{2}\left(1 - {e}^{2}\right)$

$5a = {a}^{2}\left(1 - \dfrac{4}{9}\right)$

$⇒ 5 = \dfrac{5}{9}a$

$⇒ {a}^{2} = 81, {b}^{2} = 45$

${a}^{2} + {b}^{2} = 126$.