1. The problem asks for the axis of symmetry of a parabola with vertex at the point $(5,0)$. 2. The axis of symmetry of a parabola given by a quadratic function is a vertical line that passes through the vertex. 3. The formula for the axis of symmetry is $x = h$ where $(h,k)$ is the vertex of the parabola. 4. Since the vertex is $(5,0)$, the axis of symmetry is $x = 5$. 5. Therefore, the correct answer is option C: $x = 5$.