A

Step 1: The position equations of the particle are given as:
x = a sin(kt)
y = b cos(kt)

Step 2: To find the force acting on the particle, we need to find the acceleration first. Acceleration is the second derivative of position with respect to time.
- For x: \( a_x = \frac{d^2x}{dt^2} = -a k^2 \sin(kt) \)
- For y: \( a_y = \frac{d^2y}{dt^2} = -b k^2 \cos(kt) \)

Step 3: The total acceleration \( a \) in vector form is: \( \vec{a} = a_x \hat{i} + a_y \hat{j} = -ak^2 \sin(kt) \hat{i} - bk^2 \cos(kt) \hat{j} \)

Step 4: The force is given by \( \vec{F} = m \vec{a} \), thus:
\( \vec{F} = m(-ak^2 \sin(kt) \hat{i} - bk^2 \cos(kt) \hat{j}) \)

Step 5: The magnitude of the force is:
\( |\vec{F}| = m \sqrt{(ak^2 \sin(kt))^2 + (bk^2 \cos(kt))^2} = mk^2 \sqrt{a^2 \sin^2(kt) + b^2 \cos^2(kt)} \)

The direction can be given as \( \theta = \tan^{-1} \left( \frac{a \sin(kt)}{b \cos(kt)} \right) \).
Therefore, this concludes the process for finding the force acting on the particle.