12. If the graphs of $y=x^{2}+2 a x+6 b$ and $y=x^{2}+2 b x+6 a$ intersect at only one point in the $x y$-plane, what is the $x$-coordinate of the point of intersection?
12. Answer: 3
Let $(\alpha, \beta)$ be the point of intersection of the two graphs. Then
$$
\beta=\alpha^{2}+2 a \alpha+6 b=\alpha^{2}+2 b \alpha+6 a .
$$

It follows that $2(a-b) \alpha=6(a-b)$. Since the two graphs intersect at only one point, we see that $a-b \neq 0$ (otherwise the two graphs coincide and would have infinitely many points of intersection). Consequently $2 \alpha=6$, and hence $\alpha=3$.