9. Specify all values of $a$ for which the system of equations $(x-a)^{2}=18(y-x+a-4), \log _{(x / 4)}(y / 4)=1$ has at least one solution, and solve it for each $a$.
Solution: The second equation is equivalent to the system: $x>0, x \neq 4, y=x$. Substituting $y=x$ into the first equation, we get; $(x-a)^{2}=18(a-4)$, or $x^{2}-2 a x+a^{2}-18 a+72=0(*)$, which has $D / 4=a^{2}-a^{2}+18 a-72=18(a-4)$. The number of solutions to the given system of equations depends on the number of roots of this quadratic equation.

The root $x=4$ of the quadratic equation can occur when $(4-a)^{2}=18(a-4)$, i.e., if 1) $a=4$, the equation becomes $(x-4)^{2}=0$, then this root is the only one and the given system has no solutions, or 2) $4-a=-18$, i.e., $a=22$, then for $x$ we get the equation $(x-22)^{2}=18 \cdot 18, x=22 \pm 18$, which, in addition to the extraneous root $x_{1}=4$, has another root $x_{2}=40$, satisfying the conditions, and the given system has a unique solution $(40 ; 40)$.

Consider the other cases when the system will have a unique solution.