Let $\mathbb{A}^{n}$ be the $n$-dimensional affine space over a field $k$ and $H$ an arbitrary hypersurface of $\mathbb{A}^{n}$.
Q. Does there exist a hyperplane $P$ such that $P \cap H =\emptyset$?
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Sign up to join this communityLet $\mathbb{A}^{n}$ be the $n$-dimensional affine space over a field $k$ and $H$ an arbitrary hypersurface of $\mathbb{A}^{n}$.
Q. Does there exist a hyperplane $P$ such that $P \cap H =\emptyset$?
Let me expand my comment into a (negative) answer, so the question will not appear unanswered anymore.
Let us take as $H$ the union of two non-parallel hyperplanes of $\mathbb{A}^n$. Then, clearly, there exists no hyperplane $P$ disjoint from $H$.
In fact, as remarked by Mohan, a general affine hypersurface of $\mathbb{A}^n$ of degree at least $2$ intersects every hyperplane.