We see that the vertex of the quadratic function $y=x^{2}-4x+a^{2}$ is $\left(2, a^{2}-4\right)$. If $\left(2, a^{2}-4\right)$ will be on the line $y=x+a^{2}-6$, $a^{2} -4=2+a^{2}-6$. Solve for $a$, any value of $a$ will satisfy this equation. 
For how many values of $a$ is it true that the line $y=x+a^{2}-6$ passes through the vertex of the parabola $y=x^{2}-4x+a^{2}$? (Adapted From 2005 AMC 12B Problem, Question \#8) 
  $$0$$ 
  $$1$$ 
  $$2$$ 
  $$10$$ 
  infinitely many 