Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $a_i$ is a polynomial of degree $d_i$ with coefficients in the base field $k$. We may assume that the $a_i$ are general and not all of the are constant.

Are there results giving non-trivial conditions on the $d_i$ (for instance an upper bound on $d_0+d_3+d_5$) ensuring the existence of a $k$-point on $S$?

Let $k$ be a field of characteristic zero. In the affine space $\mathbb{A}_{x,y,t}^3$ consider a surface $S$ of the form $$ S = \{a_0(t)x^2+a_1(t)xy+a_2(t)x+a_3(t)y^2+a_4(t)y+a_5(t) = 0\} $$ where $a_i$ is a polynomial of degree $d_i$ with coefficients in the base field $k$. We may assume that the $a_i$ are general. Let $P(t)$ be the determinant of the matrix $$ \left(\begin{array}{ccc} a_0(t) & \frac{a_1(t)}{2} & \frac{a_2(t)}{2} \\ \frac{a_1(t)}{2} & a_3(t) & \frac{a_4(t)}{2} \\ \frac{a_2(t)}{2} & \frac{a_4(t)}{2} & a_5(t) \end{array}\right) $$ and assume that $\deg(P) = 7$. Does $S$ have a $k$-point?