<s>[INST] Finding parameters of an ellipse in terms of Semi-Latus Rectum and Directrix. I am trying to solve for all the parameters of an ellipse in terms of the Semi-Latus Rectum, $\ell$, and Directrix, $x$. This is for some equation tables I am making so I am looking for the most simplistic expression. In nearly all the cases it comes down to solving a cubic function. 
For example in the case of the linear eccentricity, $c$, I know that:
$$
\ell=\frac{\sqrt{c}(c-x)}{\sqrt{x}}
$$
Which gets rearranged into:
$$
c^3-2c^2 x+c x^2-x \ell^2 = 0
$$
Solving for c using the normal general solution to the cubic equation gives me something nasty. Mathematica also doesn't help. Using a trigonometric solution helps however it gives me a piecewise solution:
$$
c =\begin{cases}
 &  \dfrac{2x}{3}  \left (1 + sin \left (\frac{arcsin\left (1 - \frac{27 \ell^2}{2 x^2}\right )}{3} \right ) \right )\\ 
 & \dfrac{4x}{3}  sin^2 \left (\frac{arccos\left (1 - \frac{27 \ell^2}{2 x^2}\right )}{6} \right )
\end{cases} 
$$ 
I feel that this can be simplified further through either assumptions on $x$ and $\ell$ or some trig identity I am unaware of. 
If anyone has a reference where they solved for the parameters of an ellipse in terms of just the semi-latus rectum and directrix that would be wonderful. Otherwise any insights on how to reduce the answer further or other methods to attack this problem would be appreciated.
 [/INST] There's one bug in the first equation though it's unaffected after squaring both sides.
\begin{align}
  x &= \frac{a^2}{c} \\
  a &= \sqrt{cx} \\
  \ell &= \frac{b^2}{a} \\
  &= \frac{a^2-c^2}{a} \\
  &= \frac{cx-c^2}{\sqrt{cx}} \\
  \ell \sqrt{x} &= \sqrt{c} \, \color{red}{(x-c)} \\
  0 &= c\sqrt{c}-x\sqrt{c}+\ell \sqrt{x} \\
  \sqrt{c} &= 2\sqrt{\frac{x}{3}} \sin
  \left(
     \frac{1}{3} \sin^{-1} \frac{3\sqrt{3}\, \ell}{2x}+\frac{2k\pi}{3}
  \right) \\
  c &= \frac{4x}{3} \sin^2
  \left(
    \frac{1}{3} \sin^{-1} \frac{3\sqrt{3}\, \ell}{2x}+\frac{2k\pi}{3}
  \right)
\end{align}

It is quite standard for a cubic equation.  The roots are irreducible and also  inconstructible by compasses and ruler.

Further points to be noticed:


*

*$k=0,1 \implies\sqrt{c}>0 \implies \text{two ellipses}$

*$k=2 \quad \implies\sqrt{c}<0 \implies \text{a hyperbola}$

*$\dfrac{X^2}{\frac{4x^2}{3} \sin^2
    \left(
      \frac{1}{3} \sin^{-1} \frac{3\sqrt{3}\, \ell}{2x}+\frac{2k\pi}{3}
    \right)}+
    \dfrac{Y^2}{\frac{2\ell x}{\sqrt{3}} \sin
    \left(
      \frac{1}{3} \sin^{-1} \frac{3\sqrt{3}\, \ell}{2x}+\frac{2k\pi}{3}
    \right)}=1$

*$e=\dfrac{2}{\sqrt{3}}
      \left|
        \sin \left(
               \dfrac{1}{3} \sin^{-1} \dfrac{3\sqrt{3}\, \ell}{2x}+
               \dfrac{2k\pi}{3}
             \right)
      \right|$

</s>