1. The equation of a circle passing through \((0, a)\) and \((0, -a)\) is:
\[ x^2 + y^2 + 2gx + 2fy + c = 0. \]
2. Since the circles pass through \((0, a)\) and \((0, -a)\):
\[ f = 0, \quad c = -a^2. \]
3. The equation simplifies to:
\[ x^2 + y^2 + 2gx + c = 0. \]
4. If the circles touch the line \(y = mx + c\) orthogonally, the condition for orthogonality is:
\[ c^2 = a^2(2 + m^2). \]
Thus, the correct answer is \(c^2 = a^2(2 + m^2)\).