We see that the vertex of the quadratic function $y=x^{2}-6x+a^{2}$ is $\left(3, a^{2}-9\right)$. If $\left(2, a^{2}-1\right)$ will be on the line $y=x+a^{2}-a-3$, $a^{2} -9=3+a^{2}-a-3$. Solve for $a$, there is one solution, $a=9$. 
For how many values of $a$ is it true that the line $y=x+a^{2}-a-3$ passes through the vertex of the parabola $y=x^{2}-6x+a^{2}$? (Adapted From 2005 AMC 12B Problem, Question \#8) 

1: $$0$$ 
2: $$1$$ 
3: $$2$$ 
4: $$10$$ 
5: infinitely many 