After applying a rigid motion to $S$ and $P$, we may as well assume $(x_0,y_0,z_0)=(0,0,0)$ and $P$ is the $xy$-plane. Let $U$ be a small neighborhood of the origin in $S$; by taking $U$ to be small enough, we may assume that $U$ is the image of an open disk in $\mathbb R^2$ under a local parametrization, and thus $U$ is homeomorphic to a disk. The hypothesis implies that the origin is the only point of $U$ for which the $z$-coordinate is zero, so $U\smallsetminus\{(0,0,0)\}$ is contained in the set where $z\ne 0 $. Since a disk minus a point is connected, it follows that either $z>0$ on all of $U\smallsetminus\{(0,0,0)\}$ or $z<0$ on all of $U\smallsetminus\{(0,0,0)\}$. After reflecting in the $xy$-plane, we may assume it is $z>0$.
Now suppose $v$ is a tangent vector to $S$ at the origin. Then there is a smooth curve $c:(-\varepsilon,\varepsilon)\to U$ satisfying $c(0) = (0,0,0)$ and $c'(0) = v$. If we write $c(t) = (x(t),y(t),z(t))$, we see that $z(t)$ attains a minimum at $t=0$, and therefore $z'(0)=0$. This means that every tangent vector at the origin has zero $z$-coordinate, so $T_{(0,0,0)}S$ is contained in the $xy$-plane (i.e., in $P$). Since both $T_{(0,0,0)}S$ and $P$ are two-dimensional, they must be equal.
This one
$$z-z_0 = f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)$$
is the general expression for the tangent plane to $z=f(x,y)$ at the point $(x_0,y_0,z_0)$ while
$$f_{x}(x_0)(x-x_0) +f_y(y_0)(y-y_0)=0$$
is the intersection of the tangent plane with $x-y$ plane $(z=0)$ when $z_0=0$.
For example $z=f(x,y)=x^2+y^2 \implies f_x=2x \quad f_y=2y$ then at $(1,1,2)$ the tangent plane is
$$z-2=2(x-1)+2(y-1)$$
Best Answer
You need to write equation of plane in implicit form ie $F(x,y,z) = \sqrt x + \sqrt y + \sqrt z - 1 = 0$. Then you can write equation of tangent as $F_x(x,y,z) (x-x_0) + F_y (x,y,z)(y - y_0) + F_z (x,y,z)(z-z_0) = 0 $. And finally substitute suitable zeros (eg y =0, z=0 for x-intercept) to find all 3 intercepts. When you add them it would come as some function of $(\sqrt x + \sqrt y + \sqrt z)$, hence constant.