Show that the length of the polar subtangent

Show that the length of the polar subtangent of the hyperbolic spiral $\rho=a/\varphi$ is constant.

Step 1: The polar subtangent

For $\rho=f(\varphi)$, the length of the polar subtangent is $S_T=\left|\rho^2/\rho'(\varphi)\right|$.

Step 2: Differentiate

\[ \frac{d\rho}{d\varphi} = -\frac{a}{\varphi^2}, \qquad \rho^2 = \frac{a^2}{\varphi^2}. \tag{1} \]

Step 3: Compute $S_T$

\[ S_T = \left|\frac{a^2/\varphi^2}{-a/\varphi^2}\right| = |-a| = a. \tag{2} \]

Step 4: Verification

\[a=6,\ \varphi=0.5,\,2.0,\,5.0\text{: } S_T=\rho^2/|\rho'|=6.00000\text{ in every case}\ \checkmark\]