 Equality of horizontal hyperbolic and Euclidean distances

For points $ib$ and $a+ib$ with $a,b>0$, the hyperbolic distance is $2\mathrm{arsinh}(a/(2b))$. It equals their Euclidean distance $a$ exactly when the displayed relation holds. The function of $a$ decreases continuously from one to zero: its derivative has the sign of $\sinh (a/2)-(a/2)\cosh (a/2)<0$. Hence every height $b\in (0,1)$ has exactly one positive matching horizontal separation.