1. Find the midpoint of AB:

\(\left( \frac{1+9}{2}, \frac{3+(-1)}{2} \right) = (5, 1)\)

2. Calculate the slope of AB:

\(m_{AB} = \frac{-1 - 3}{9 - 1} = \frac{-4}{8} = -\frac{1}{2}\)

3. The slope of the perpendicular bisector is the negative reciprocal:

\(m = 2\)

4. Use the point-slope form to find the equation of the perpendicular bisector:

\(y - 1 = 2(x - 5)\)

5. Simplify to find the equation:

\(y = 2x - 9\)

6. Set \(x = 0\) to find the intersection with the y-axis:

\(y = 2(0) - 9 = -9\)

7. Thus, the coordinates of C are \((0, -9)\).