A

Step 1: Given the trajectory of the particle is a parabola defined by the equation
$$ y = \frac{1}{4}x^2 $$
We first find the first derivative to determine the slope of the tangent.
Step 2: Differentiate y with respect to x:
$$ \frac{dy}{dx} = \frac{1}{2}x $$
Step 3: Now compute the second derivative to find the curvature:
$$ \frac{d^2y}{dx^2} = \frac{1}{2} $$
Step 4: The radius of curvature (R) can be calculated using the formula:
$$ R = \frac{[1 + (\frac{dy}{dx})^2]^{3/2}}{|\frac{d^2y}{dx^2}|} $$
Step 5: At the point where x = 0:
$$ \frac{dy}{dx} = \frac{1}{2}(0) = 0 \quad \text{and} \quad \frac{d^2y}{dx^2} = \frac{1}{2} $$
Step 6: Substitute these into the radius of curvature formula:
$$ R = \frac{[1 + 0^2]^{3/2}}{|\frac{1}{2}|} = \frac{1}{\frac{1}{2}} = 2 $$
Therefore, the radius of curvature at point x = 0 is 2.