$\newcommand{\angles}[1]{\left\langle\,{#1}\,\right\rangle}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
\newcommand{\half}{{1 \over 2}}
\newcommand{\ic}{\mathrm{i}}
\newcommand{\iff}{\Longleftrightarrow}
\newcommand{\imp}{\Longrightarrow}
\newcommand{\Li}[1]{\,\mathrm{Li}_{#1}}
\newcommand{\mc}[1]{\mathcal{#1}}
\newcommand{\mrm}[1]{\mathrm{#1}}
\newcommand{\ol}[1]{\overline{#1}}
\newcommand{\pars}[1]{\left(\,{#1}\,\right)}
\newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
\newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
\newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
\newcommand{\ul}[1]{\underline{#1}}
\newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
The velocities $\ds{\vec{v}_{i}\pars{t}\,,\ \pars{~i = A, B~}}$, satisfy
$\ds{t}$-independent $\ds{v_{i} \equiv \verts{\vec{v}_{i}\pars{t}} \equiv v}$.
$\ds{v}$ is the common rapidity.
\begin{align}
\vec{v}_{B}\pars{t} & =
{\vec{r}_{A}\pars{t} - \vec{r}_{B}\pars{t} \over
\verts{\vec{r}_{A}\pars{t} - \vec{r}_{B}\pars{t}}}\,v\,,\qquad
\vec{r}_{A}\pars{t} = vt\,\hat{x}
\end{align}
Lets $\ds{\vec{r}_{B}\pars{t} \equiv
\vec{r}_{A}\pars{t} + \rho\pars{t}\cos\pars{\theta\pars{t}}\,\hat{x} + \rho\pars{t}\sin\pars{\theta\pars{t}}\,\hat{y}}$:
$$
\imp\quad
\bracks{v + \cos\pars{\theta}\dot{\rho} - \rho\sin\pars{\theta}\dot{\theta}}\hat{x}
+\bracks{\sin\pars{\theta}\dot{\rho} + \rho\cos\pars{\theta}\dot{\theta}}\hat{y} =
-v\cos\pars{\theta}\hat{x} - v\sin\pars{\theta}\hat{y}
$$
\begin{align}
&\left\lbrace\begin{array}{rcrcl}
\cos\pars{\theta}\dot{\rho} & - & \rho\sin\pars{\theta}\dot{\theta} & = &
-v\cos\pars{\theta} - v
\\[2mm]
\sin\pars{\theta}\dot{\rho} & + & \rho\cos\pars{\theta}\dot{\theta} & = &
-v\sin\pars{\theta}
\end{array}\right.
\\[5mm]
\imp\quad &
\left\lbrace\begin{array}{rcl}
\dot{\rho} & = &\bracks{-v\cos\pars{\theta} - v}\rho\cos\pars{\theta} -
\bracks{-v\sin\pars{\theta}}\bracks{-\rho\sin\pars{\theta}} =
\bracks{-v - v\cos\pars{\theta}}\rho
\\[2mm]
\dot{\theta} & = &\cos\pars{\theta}\bracks{-v\sin\pars{\theta}} -
\sin\pars{\theta}\bracks{-v\cos\pars{\theta} - v} = v\sin\pars{\theta}
\end{array}\right.
\end{align}
\begin{align}
{1 \over \rho}\,\totald{\rho}{\theta} & =
-\,{1 + \cos\pars{\theta} \over \sin\pars{\theta}}
\\[5mm]
\ln\pars{\rho_{\infty} \over k} & =
\left.-2\ln\pars{\sin\pars{\theta \over 2}}\,\right\vert_{\pi/2}^{\pi} =
\ln\pars{\half}\quad\imp\quad
\color{#f00}{\rho_{\infty}} = \color{#f00}{k \over 2}
\end{align}
