Is a binary quadratic form (over any field) that represents both $\pm 1$ necessarily hyperbolic?

Question

If a $2$-dimensional quadratic form over a field $\mathbb F$ that represents both $1$ and $-1$ necessarily hyperbolic?

Edit: Assume that $\text{char } \mathbb{F} \neq 2$.

Answer 1

The answer is no: In any field in which $-1$ is a square (e.g., for any field of characteristic $2$ or characteristic $p$, $p \equiv 1 \bmod 4$, or any field that contains $\mathbb{Q}[i]$), say, $-1 = f^2$, the quadratic form $Q((x, y)) := x^2$ satisfies $Q((1, 0)) = 1$ and $Q((f, 0)) = -1$ but again the quadratic form is degenerate.