1. * Given the element $(1,2) \in A \cap B$, where $A=\left\{(x, y) \mid a x-y^{2}+b=0\right\}, B=$ $\left\{(x, y) \mid x^{2}-a y-b=0\right\}$. Find the values of $a, b$.
From the conditions, we have 
\[
\left\{\begin{array}{l}
a \cdot 1-2^{2}+b=0, \\ 
1^{2}-a \cdot 2-b=0,
\end{array}\right.
\]
solving these, we get \(a=-3, b=7\).