I have a field $K$ of transcendence degree two over $\mathbb{R}$, and elements $a_1,a_2,a_3\in K$. I would like to understand the set $$ Q = \{ u\in K^3 : \sum_i a_iu_i^2 = 1\} $$ In particular, I would like to know whether it is nonempty, and if so, I would like to find some examples of elements, and ideally some kind of formula for all elements. What general methods are known for this sort of question? My gut feeling is that $Q$ is likely to be empty for most choices of the parameters $a_i$; are there any theorems of that type?

Specifically, I am interested in the following case: \begin{align*} r &= y_2^3+(5y_1-9)y_2^2/5+(-20y_1+4y_1^2+24-16y_4)y_2/25-4(1-y_1)/25 \\ A &= \mathbb{R}[y_1,y_4,y_2]/(r) \\ K &= \text{field of fractions of } A \\ a_1 &= y_2(1-y_1-y_2) \\ a_2 &= y_4 \\ a_3 &= y_2. \end{align*}

Note that $A$ is a free module of rank three over the subring $A_1=\mathbb{R}[y_1,y_4]$, so $K$ is a free module of rank three over the purely transcendental subfield $K_1=\mathbb{R}(y_1,y_4)$. Using this, we could translate the question to one about a quadratic form in nine variables over $K_1$, which might or might not be more tractable.