To determine the locus of the center of a circle passing through the point (a, 0) and touching the line x=1, we can use the concept of the perpendicular distance from a point to a line.

Let's analyze the problem step by step:

1. Definition of locus:
The locus is the set of all points that satisfy a given condition or set of conditions. In this case, we are looking for the locus of the center of the circle.

2. Equation of the circle:
A circle is defined by its center (h, k) and radius r. The equation of a circle can be written as:
(x - h)^2 + (y - k)^2 = r^2

3. Center of the circle:
The center of the circle will have coordinates (h, k).

4. Circle passing through (a, 0):
Since the circle passes through the point (a, 0), we can substitute these coordinates into the equation of the circle:
(a - h)^2 + (0 - k)^2 = r^2

5. Circle touching the line x=1:
The line x=1 is a vertical line. The distance between the center of the circle and the line x=1 must be equal to the radius of the circle for it to touch the line. The distance between a point (x, y) and the line x=1 is given by |x-1|.

6. Locus condition:
Therefore, the condition for the circle to touch the line x=1 can be expressed as:
|h - 1| = r

7. Substituting the equation of the circle and the locus condition:
Substituting the equation of the circle and the locus condition into each other, we get:
(a - h)^2 + (0 - k)^2 = (h - 1)^2

8. Simplifying the equation:
Expanding and simplifying the equation, we have:
a^2 - 2ah + h^2 + k^2 = h^2 - 2h + 1

9. Simplifying further:
Canceling out the h^2 terms, we get:
a^2 - 2ah + k^2 = 1 - 2h

10. Rearranging the equation:
Rearranging the equation, we have:
k^2 - 2ah + 2h - a^2 + 1 = 0

11. Comparing with the equation of a parabola:
Comparing the equation with the standard equation of a parabola (y^2 = 4ax), we can see that it matches. Therefore, the locus of the center of the circle is a parabola.

Hence, the correct answer is option C) Parabola.