Let $I\subset k [x_0,\dots, x_n] $ be a homogeneous prime ideal, and denote by $\bar {I} $ the ideal of $k(u_0,\dots,u_n) [x_0,\dots, x_n] $ generated by $I $ and the element $x_0u_0+\dots+x_nu_n $.
Assume that $k $ is an infinite field. Is it true that $\bar {I} $ is a prime ideal?
The answer is yes. Let me do this for projective varieites (since $I$ is a homogeneous ideal). So, $I$ defines an irreducible variety $X\subset \mathbb{P}^n$ and you have a subscheme $\Gamma$ defined by your equation in $X\times\mathbb{P}^n$. The projection map $\Gamma\to X$ has all fibers just $\mathbb{P}^{n-1}$ and since $X$ is irreducible, this shows that $\Gamma$ is an irreducible variety (If you have never seen a proof of this, I suggest looking it up in a book, say Shafarevich's Algebraic Geometry). The scheme you are interested in is the the fiber over the generic point of the projection $\Gamma\to \mathbb{P}^n$ and clearly this is irreducible since $\Gamma$ itself is.
Similar argument will also work for the affine case that you are interested in.