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

The root $x=2$ of the quadratic equation can occur when $(2-a)^{2}=9(a-2)$, i.e., if 1) $a=2$, the equation becomes $(x-2)^{2}=0$, then this root is the only one and the given system has no solutions, or 2) $2-a=-9$, i.e., $a=11$, then for $x$ we get an equation that, in addition to the extraneous root $x_{1}=2$, has another root $x_{2}=20$, satisfying the conditions, and the given system has a unique solution ( $20 ; 20$ ).

Let's consider the other cases when the system will have a unique solution.