11. Determine the largest value of $a$ such that $a$ satisfies the equations $a^{2}-b c-8 a+7=0$ and $b^{2}+c^{2}+b c-6 a+6=0$ for some real numbers $b$ and $c$.
11. Ans: 9
Substituting the first equation $b c=a^{2}-8 a+7$ into the second equation. we have $(b+c)^{2}=(a-1)^{2}$ so that $b+c= \pm(a-1)$. That means $b$ and $c$ are roots of the quadratic equation $x^{2} \mp(a-1) x+\left(a^{2}-8 a+7\right)=0$. Thus its discriminant $\Delta=[\mp(a-1)]^{2}-4\left(a^{2}-8 a+7\right) \geq 0$, or equivalently, $1 \leq a \leq 9$. For $b=c=4$, $a=9$ satisfies the two equations. Thus the largest value of $a$ is 9 .