Tangent cone of an arbitrary algebraic curve

Question

So, my problem is: given a real/complex (I will assume complex) algebraic curve, say $f(x,y)$, or $f(z,w)$ for $x,y\in\mathbb{R}$ or $z,w\in\mathbb{C}$ (I would like to hear your thoughts in either case), and let's assume that $\frac{\partial f}{\partial x}(x_0,y_0)=0$, and $\frac{\partial f}{\partial y}(x_0,y_0)=0$ at the point $(x_0,y_0)$; in other words, $(x_0,y_0)$ is defining a singularity, assuming $(x_0,y_0)$ lies on the locus of $f$.

I want to find the equations for the tangent lines at $(x_0,y_0)$. A few examples I am interested in are:

1.) $f=xy+x+y+1$

2.) $f=x^3-x^2+y^2$

3.) $f=x^3+x^2+y^2$

4.) $f=2x^4-3x^2y+y^2-2y^3+y^4$

I read about a definition here: Tangents at singularities, but was unable to figure out how to use it computationally. Additionally, I read about a "minimal homogeneous degree" here: Compute tangent cone from a given ideal?, but was unable to figure out what was meant, particularly in case "1.)"

I would appreciate any references, thoughts, and or ideas that you have! Ultimately, I am trying to teach myself about the classification of singular points. I have been using "Walker, Robert John. Algebraic Curves. Dover, 1962." along with other papers I have found on the internet and the maple "algcurves" package and the "singularities" command in it. Here are some links for those:

https://www.maplesoft.com/support/help/maple/view.aspx?path=algcurves%2Fsingularities,

https://www.math.fsu.edu/~hoeij/algcurves.html

Answer 1

1. $(-1,-1)$ $$(x+1)(y+1)=0$$

2. $(0,0)$ $$y^2-x^2=(y+x)(y-x)=0$$

3. $(0,0)$ $$y^2+x^2=(y+ix)(y-ix)=0$$

4. $(0,0)$ $$2x^4-3x^2y+y^2-2y^3+y^4=0$$

$$y^2=0$$

4. $(0,1)$ $$(y-1)^4+2(y-1)^3+(y-1)^2-3x^2(y-1)+2x^4-3x^2=0$$

$$(y-1)^2-3x^2=0$$