Given that the length of the latus rectum of the ellipse is 10, we have:

Next, consider the function . To find its minimum value, we calculate the derivative:

Plugging into gives the minimum value:

Thus, the eccentricity of the ellipse is , so .
Using the eccentricity formula for an ellipse:

Rearranging gives:

From equation (1), . Substituting, we have:

Solving for , and therefore

Finally, calculate :

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