Conic Sections · Ellipse latus rectum and eccentricity

Let the length of the latus rectum of an ellipse

\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1

(a > b)

, be 30. If its eccentricity is the maximum value of the function

f(t) = -\dfrac34 + 2t - t^2

, then

(a^2 + b^2)

is equal to

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