A question relating to the "hanging cable problem" in which a cable hangs from two poles in the form of a catenary. Typically the problem is to solve for the sag or distance between poles given the length of the cable. My problem is different; given known equal height poles and known distance between them, with the cable tangent to the ground: can you solve for the scaling factor a?
$$y = a \cdot cosh \left(\frac{x}{a} \right) - a$$
Assuming that I properly understood, knowing $x$ and $y$, you want to solve for $a$ the equation $$y = a \, \cosh \left(\frac{x}{a} \right) - a$$ Let $a=\frac x z$, $k=\frac y x$ to make $$k=\frac{\cosh (z)-1}{z}$$ To have an approximation using Taylor series $$k=\frac{z}{2}+\frac{z^3}{24}+\frac{z^5}{720}+\frac{z^7}{40320}+\frac{z^9}{3628800} +O\left(z^{11}\right)$$ Now, using series reversion $$z=2 k-\frac{2 k^3}{3}+\frac{26 k^5}{45}-\frac{622 k^7}{945}+\frac{4042 k^9}{4725}+O\left(k^{11}\right)$$