<s>[INST] Envelope of isoperimetric ellipses Find envelope of ellipses $ \frac{x^2}{a^2} + \frac{y}{b^2}=1 $ which have $a,b$ and its associated eccentricity $e$ variable while holding its perimeter
$$ p= 4 a E(e) $$
as constant.
Expected to be asteroid-like, passing through 
$$
\begin{pmatrix} 0 \\ \frac{\pi a}{2} \end{pmatrix},
\begin{pmatrix} \frac{a}{\sqrt{2}} \\ \frac{a}{\sqrt{2}} \end{pmatrix},
\begin{pmatrix} \frac{\pi a}{2} \\0 \end{pmatrix}$$
 [/INST] \begin{align*}
  p &= 4a E(k) \\
  a(k) &= \frac{p}{4E(k)} \\
  F(x,y,k) &=
  x^2+\frac{y^2}{1-k^2}-\frac{p^2}{16E^2(k)} \\
  \frac{\partial F}{\partial k} &=
  \frac{2ky^2}{(1-k^2)^2}-
  \frac{p^2}{8E^3(k)} \left[ \frac{K(k)-E(k)}{k} \right]
\end{align*}
The envelope is given by $$F=\dfrac{\partial F}{\partial k}=0$$
On solving,
$$
\begin{pmatrix}
  x \\ y
\end{pmatrix}=
\frac{p}{4kE(k)}
\begin{pmatrix}
  \pm \sqrt{1-\dfrac{(1-k^2)K(k)}{E(k)}} \; \\[5pt]
  \pm (1-k^2)\sqrt{\dfrac{K(k)}{E(k)}-1} \;
\end{pmatrix} \, , \quad 0<k<1 \tag{1}
$$
If we adimit $a<b$, then
$$
\begin{pmatrix}
  x \\ y
\end{pmatrix}=
\frac{p}{4kE(k)}
\begin{pmatrix}
  \pm (1-k^2)\sqrt{\dfrac{K(k)}{E(k)}-1} \; \\[5pt]
  \pm \sqrt{1-\dfrac{(1-k^2)K(k)}{E(k)}} \;
\end{pmatrix} \, , \quad 0<k<1 \tag{2}
$$

A plot of the envelope for $p=4$ with the ellipses is shown below:


</s>