A curve $y(x)$ in the $x y$-plane connects the points $(\pm a, 0)$ and has a fixed length $l, 2 a<l<\pi a$. Find an expression for the area $A$ of the surface of the revolution obtained by rotating $y(x)$ about the $x$-axis.

Show that the area $A$ has a stationary value for

$$y=\frac{1}{k}(\cosh k x-\cosh k a),$$

where $k$ is a constant such that

$$l k=2 \sinh k a .$$

Show that the latter equation admits a unique positive solution for $k$.