For every $a \in \mathbb{R}$, let $P_a$ be the parabola given by $y=x^2+2 a x+a$. Then

• Correct answer
  A

  there is a common point which lies on $P_a$ for each $a$
• B

  there is no common point which lies on $P_a$ for each $a$
• Correct answer
  C

  the vertex of the parabola $P_a$ lies on the line $x=-a$
• Correct answer
  D

  the locus of the vertex of $P_a$ is a parabola whose vertex is $\left(-\frac{1}{2}, \frac{1}{4}\right)$

Hence, the correct answer is options: 'A', 'C', 'D'.