Let $f \in \mathbb{C}[[x,y]]$ be the germ of an isolated plane curve singularity. Then the Hilbert-Samuel multiplicity $e_f$ of $f$ is given as follows: $$e_f = \lim_{s \to \infty}\frac{1}{s} \cdot \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)^s\right)$$ I've noticed that in various examples (like the $A_{n-1}$-singularities $f = y^2 - x^n$, or more generally, the hypercuspidal singularities $f = y^a - x^b$) that $e_f$ is equal to the intersection multiplicity of $f$ and a ``generic polar'' of $f$. More precisely, I've observed that the following equality holds in many examples: $$e_f = \dim_{\mathbb{C}} \mathbb{C}[[x,y]]\bigg/\left(f, \alpha \cdot \frac{ \partial f}{\partial x} - \beta \cdot \frac{\partial f}{ \partial y}\right) \qquad (*)$$ where $[\alpha : \beta] \in \mathbb{P}_{\mathbb{C}}^1$ is generic. My question: is it well-known (or obvious) whether the equality $(*)$ holds for an arbitrary isolated plane curve singularity germ $f$?

(Note: the colength of the polar (the quantity on the right of $(*)$) is of interest because it is related to Teissier's notion of ``polar invariant.'')

What I have so far: Note that the polar is itself a complete intersection, so the colength of the polar in the ring $\mathbb{C}[[x,y]]/(f)$ is equal to the Hilbert-Samuel multiplicity of the polar. Since the Hilbert-Samuel multiplicity of an ideal is equal to that of any reduction, it suffices to show that the ideal $I = \left(\alpha \cdot \frac{\partial f}{\partial x} - \beta \cdot \frac{\partial f}{\partial y}\right)$ generated by the polar is a reduction of the ideal $J = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$ (here, we are regarding $I,J$ as ideals of the ring $\mathbb{C}[[x,y]]/(f)$). I.e., it suffices to show that for some $s \geq 0$, the obvious inclusion $$I(J^s) \subset J^{s+1}$$ of ideals of $\mathbb{C}[[x,y]]/(f)$ is in fact an equality. Now restrict to the case where $f \in \mathbb{C}[x,y]$ is homogeneous of degree $d$. Then it is not hard to check by counting generators that $IJ^{d-1} = (x,y)^{d(d-1)} \supset J^d$ (as ideals of $\mathbb{C}[[x,y]]/(f)$). I do not currently know how to generalize this argument to handle germs $f$ that are not homogeneous.