multiple tangent lines to a plane curve Suppose that $C=V(F)$, with $F\in k[X,Y]$ ($k$ algebraically closed), is an irreducible plane curve such that $P=(0,0)$ is a nodal singular point (or an ordinary multiple point according to Fulton's book) of multiplicity $m\ge 2$. A procedure to find exactly $m$ distinct tangent lines at $P$ is the following:

If $F_m$ is the homogeneous part of degree $m$ of $F$, we factorize it in exactly $m$ linear terms:
$$F_m=\lambda\prod_{i=1}^m(X-\alpha_iY)\quad\quad  \lambda,\alpha_i\in k$$
Now each $X-\alpha_iY=0$ is a tangent line in $P$.

I don't understand the geometrical meaning of the above procedure. Why by considering the factorization of the homogeneous part of degree $m$ we obtain tangent lines? If $P$ is a non-singular point, the tangent line in $P$ is "justified" by the differential calculus, in fact  the gradient of $F$  is a tangent vector in $P$ and it spans the tangent line. So when all partial derivatives vanish in $P$, where is the geometrical/analytic insight?
Thanks in advance.
 A: Let $k=\mathbb{C}.$ I think, the best intuitive explanation is related to the notion of "tangent cone" (See Gathmann's notes page 61) which is the "best" approximation of $C$ around a singularity.
You know that the linear part of $F$ is the simplest approximation of $C$ around $P,$ so in the absence of the linear part (singularity), we should stick to the second option which is the non-zero homogeneous part $F_m$ of $F$ with the smallest degree in the homogeneous decomposition of $F.$ (for example, let $F(X,Y)=(Y^2-X^2)-X^3$ then $F_0=F_1=0,$ $F_2=Y^2-X^2$ and $F_3=-X^3.$) 
The affine algebraic set generated by the ideal $(F_m)$ forms a cone over the singularity $P$ and is called the tangent cone. (in the above example, it is $V(Y^2-X^2)\subset \mathbb{A}^2$ which contains two tangent lines $Y=\pm X$ of $C$ at $P.$) Then, if $F_m$ factorizes into linear products $\lambda\prod_{i=1}^m(X-\alpha_iY)$ the affine zero locus is the tangent cone at $P$ contianing $m$ tangent lines of $C$ at $P.$ Of course, since $\alpha_i$'s may not be distinct in general, you may have a sort of thickened tanget lines (for example, $F(X,Y)=Y^3-X^2$ then $F_2=-X^2$ and $V(X^2)$ is the thickened $x$-axis.)
Furthermore, the projectivized tangent cone i.e. $V_p(F_m)$ where $V_p(I)$ is the projective zero locus of $I$ is the exceptional locus in the blowup of $C$ at $P.$ (in our example, then the exceptional locus of $\tilde{C}$ the blowup of $C$ at $P,$ has two points corresponding to two tangent lines $Y=\pm X.$ ) 
