Is there any intuitive meaning to $\beta^T A \beta=1$? In a statistical paper, I found that $\beta^T A \beta=1$, where $\beta$ and $A$ are a vector and matrix of constants, respectively.
In the paper, the author utilizes only the $\beta$ that satisfies $\beta^T A \beta=1$, so I guess that the equation stating "a quadratic form is one" has something special meaning.
Is there any special meaning of that? In my rough idea, the quadratic form looks like theequation of a circle.
(I intentionally omitted the exact context, because they are redundant for my question)

Note that, for any vectors $a$ and $b$ in an inner-product space such as $\mathbb R^d$, one possible inner-product can be defined as $$a^T b = a \cdot b = \|a\|\|b\| \cos \theta$$ where $\|\cdot\|$ is the Euclidean norm of the vector and $\theta$ is the angle between the two vectors. Therefore,
\begin{align}
\beta^T A\beta &= \beta^T(A\beta) \\
&= \beta \cdot (A\beta) \\
&= \|\beta\| \|A\beta \| \cos \theta
\end{align}
where $\theta$ now represents the angle between $\beta$ and $A\beta$. Then,
\begin{align}
\beta^T A\beta &= 1 \\
\|\beta\| \|A\beta \| \cos \theta &= 1 \\
\theta &= \arccos \left(\frac{1}{\|\beta\| \|A\beta \|}\right)
\end{align}
Hence, we are interested in the vectors $\beta$ such that the angle between $\beta$ and $A\beta$ is $\arccos \left(\frac{1}{\|\beta\| \|A\beta \|}\right)$.
More formally, let $A$ be the matrix representation of the linear operator $T : \mathbb R^d \to \mathbb R^d$ in the inner-product space $\mathbb R^d$ defined with the same inner product above. We are interested in the set of vectors $\beta$ such that $\beta$ and $T(\beta)$ have an angle $\theta$ between them.
