Let $k$ be a field of characteristic $\neq 2$ and $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle$ a Pfister form over $k$. Denote by $Q_{\underline{a}}=Q_{a_{1},\cdots,a_{n}}$ the projective quadric of dimension $2^{n-1}-1$ given by the equation $\langle\!\langle a_{1},\cdots,a_{n-1}\rangle\!\rangle=a_{n}t^{2}$. Let $P_{\underline{a}}$ denote the quadric given by the equation $\langle\!\langle a_{1},\cdots,a_{n}\rangle\!\rangle=0$.

Question: If $P_{\underline{a}}$ has a $k$-rational point then does $Q_{\underline{a}}$ have a $k$-rational point?

This question is Lemma 4.2 in “Motivic cohomology with $\mathbb{Z}/2$-coefficients” by V. Voevodsky. I do not understand this proof; for any rational point $p$ of $P_{\underline{a}}$, why can he say that there exists a linear subspace $H$ of dimension $2^{n-1}-1$ which lies on $P_{\underline{a}}$ and passes through $p$?


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