Ratio of Displacements in Oscillatory System

To determine the ratio $x_1 / x_2$, we analyze the system executing simple harmonic motion (SHM).

1. Energy in Simple Harmonic Motion:
   For a particle of mass $m$ executing simple harmonic motion with amplitude $A$ and angular frequency $\omega$, the total mechanical energy of the system is given by:
   $E = \frac{1}{2} k A^2$
   where $k = m\omega^2$ is the effective spring constant.

2. Condition for $x_1$:
   The displacement $x_1$ corresponds to the position where the kinetic energy ($K$) is equal to the potential energy ($U$):
   $K(x_1) = U(x_1) = \frac{1}{2} E$
   Since the potential energy at any displacement $x$ is $U(x) = \frac{1}{2} k x^2$, we have:
   $\frac{1}{2} k x_1^2 = \frac{1}{2} \left(\frac{1}{2} k A^2\right)$
   $x_1^2 = \frac{A^2}{2} \implies x_1 = \frac{A}{\sqrt{2}}$

3. Condition for $x_2$:
   The displacement $x_2$ corresponds to the position where the potential energy is one-fourth of the total energy, or where the kinetic energy is three times the potential energy ($K = 3U$):
   $U(x_2) = \frac{1}{4} E$
   Substituting $U(x_2) = \frac{1}{2} k x_2^2$:
   $\frac{1}{2} k x_2^2 = \frac{1}{4} \left(\frac{1}{2} k A^2\right)$
   $x_2^2 = \frac{A^2}{4} \implies x_2 = \frac{A}{2}$

4. Ratio of Displacements:
   Taking the ratio of $x_1$ to $x_2$:
   $\frac{x_1}{x_2} = \frac{\frac{A}{\sqrt{2}}}{\frac{A}{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}$

Therefore, the correct option is (C).