The two circles x² + y² = ax and x² + y² = c²(c > 0) touch each other if :-

Let c1, c2 be the centers of the two circles. c1 = (a/2, 0), c2 = (0, 0). r1 = |a|/2, r2 = c.
For the circles to touch each other, the distance between their centers must be equal to the difference or sum of their radii. Since c > 0, we consider the case where they touch externally:
c1c2 = r1 + r2
√((a/2 - 0)² + (0 - 0)²) = |a|/2 + c
|a|/2 = |a|/2 + c
This equation has no solution unless a = 0, which is not consistent with the problem statement. Let's consider the case where the circles touch internally:
c1c2 = |r1 - r2|
√((a/2 - 0)² + (0 - 0)²) = ||a|/2 - c|
|a|/2 = ||a|/2 - c|
This leads to two possibilities:

1. |a|/2 = |a|/2 - c => c = 0, which contradicts c > 0.
2. |a|/2 = c - |a|/2 => |a| = c
   Hence, |a| = c. Therefore, option C is correct.