What do you mean by the "envelope" of the $\mathcal{E}_1$ set of ellipses?
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[![Ellipses][1]][1]
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$\mathcal{E}_1$: $a=1$; $e=0,0.1,\ldots,0.9,1$.
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  [1]: https://i.sstatic.net/yjrU7.jpg