Consider the projective curve $C=V(P)$ in $\mathbb{P}^2$ where $P(x_0,x_1,x_2)$ is an homogeneous polynomial of degree $d$. At a point $[a,b,1]$, the multiplicity of $C$ is
$$Mult_{[a,b,1]}C=max\left \{k:\frac{\partial^{\alpha}P}{\partial{x_0}^{\alpha_0}\partial{x_1}^{\alpha_1}\partial{x_2}^{\alpha_2}}[a,b,1] \; \; ,\forall\alpha\in\mathbb{N}^3 \; \mbox{with}\,\,|\alpha|<k\right \}$$.
Let $f(x,y)=P(x,y,1)$. This defines an affine curve $D$ in $\mathbb{C}^2$. Define its multiplicity at a point $(a,b)$ using an analogue of the definition above (but clearly with $\alpha\in\mathbb{N}^2$.
How can I argue that $Mult_{[a,b,1]}C=Mult_{(a,b)}D$?