If we plot the graph of any $z=f(x,y)$ we will get a surface. If we take the partial derivatives at $(x_0,y_0)$ we will have two partial derivatives $f_x$ and $f_y$.
The equation of the tangent lines at that point along the $x$-axis and $y$-axis will be, respectively, $$z=z_0+f_x(x-x_0),y=y_0$$
As far I know the parametric forms of the two lines will be $$(x,y,z)=(x_0+t_1,y_0,z_0+f_xt_1)$$ and $$(x,y,z)=(x_0,y_0+t_2,z_0+f_yt_2)$$
My question is, am I right? If I'm not, what is correct parametric form of the two equations? I also want to know the vector and symmetric forms of the two equations with an explanation.