For how many values of $ a$ is it true that the line $ y \equal{} x \plus{} a$ passes through the vertex of the parabola $ y \equal{} x^2 \plus{} a^2$?

$ \textbf{(A)}\ 0\qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ 2\qquad \textbf{(D)}\ 10\qquad \textbf{(E)}\ \text{infinitely many}$
1. First, identify the vertex of the parabola \( y = x^2 + a^2 \). The standard form of a parabola \( y = ax^2 + bx + c \) has its vertex at \( x = -\frac{b}{2a} \). In this case, \( a = 1 \), \( b = 0 \), and \( c = a^2 \), so the vertex is at \( x = 0 \). Substituting \( x = 0 \) into the equation of the parabola, we get:
   \[
   y = 0^2 + a^2 = a^2
   \]
   Therefore, the vertex of the parabola is at the point \( (0, a^2) \).

2. Next, consider the line \( y = x + a \). This line passes through the y-axis at the point \( (0, a) \).

3. For the line \( y = x + a \) to pass through the vertex of the parabola \( (0, a^2) \), the y-coordinate of the vertex must satisfy the equation of the line when \( x = 0 \). Therefore, we set the y-coordinate of the vertex equal to the y-intercept of the line:
   \[
   a^2 = a
   \]

4. Solve the equation \( a^2 = a \):
   \[
   a^2 - a = 0
   \]
   Factor the quadratic equation:
   \[
   a(a - 1) = 0
   \]
   This gives us two solutions:
   \[
   a = 0 \quad \text{or} \quad a = 1
   \]

5. Therefore, there are two values of \( a \) that satisfy the condition that the line \( y = x + a \) passes through the vertex of the parabola \( y = x^2 + a^2 \).

The final answer is \(\boxed{2}\)