Let $d_{0}>0$, $R>0$ and $p_{3}\in \left( 0,d_{0}\right) $ given. Consider $C_{c}$ a any catenoid of radius $c>0$ centered $p=\left( 0,0,p_{3}\right)$ in $% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ^{3}$. Let $B$ a closed ball in the $z=0$ plane centered on the origin of radius $R$.
Show that there $c\in \left[ 0,R\right] $ such that $C_{c}$ is centered in $p=\left( 0,0,p_{3}\right) $ and
$$% C_{c}\cap \left\{ z=0\right\} =\partial B$$
The problem is very intuitive. I tried to use the equation that describes the points of a catenoid, but I could not solve the problem, $% R=c\cosh\left( \frac{p_{3}}{c}\right) $.