Step-by-Step Solution:

1. Principle of Energy Conservation The problem states that the entire potential energy of a compressed spring is used to impart angular velocity to each disc. This is a direct application of the principle of conservation of energy, where the initial potential energy of the spring is converted into the final rotational kinetic energy of the disc.

2. Formulas for Energy

   • The potential energy ($U$) stored in a spring with spring constant $k$ compressed by a distance $x$ is given by: $U = {1 \over 2} k x^2$
   • The rotational kinetic energy ($K_{rot}$) of a body with moment of inertia $I$ rotating with angular velocity $\omega$ is given by: $K_{rot} = {1 \over 2} I \omega^2$

3. Analysis for Disc A

   • Moment of inertia of disc A, $I_A = I$.
   • Angular velocity of disc A, $\omega_A = 2\omega$.
   • The spring is compressed by a distance $x_1$.
   • According to the conservation of energy: Potential Energy of spring 1 = Rotational Kinetic Energy of Disc A $U_1 = K_A$ ${1 \over 2} k x_1^2 = {1 \over 2} I_A \omega_A^2$
   • Substituting the given values: ${1 \over 2} k x_1^2 = {1 \over 2} I (2\omega)^2$ ${1 \over 2} k x_1^2 = {1 \over 2} I (4\omega^2)$ ${1 \over 2} k x_1^2 = 2 I \omega^2 \quad \cdots(1)$

4. Analysis for Disc B

   • Moment of inertia of disc B, $I_B = 2I$.
   • Angular velocity of disc B, $\omega_B = \omega$.
   • The spring used has the same spring constant $k$ and is compressed by a distance $x_2$.
   • According to the conservation of energy: Potential Energy of spring 2 = Rotational Kinetic Energy of Disc B $U_2 = K_B$ ${1 \over 2} k x_2^2 = {1 \over 2} I_B \omega_B^2$
   • Substituting the given values: ${1 \over 2} k x_2^2 = {1 \over 2} (2I) (\omega)^2$ ${1 \over 2} k x_2^2 = I \omega^2 \quad \cdots(2)$

5. Calculate the Ratio $x_1 / x_2$

   • To find the ratio, we can divide Equation (1) by Equation (2): ${{{1 \over 2} k x_1^2} \over {{1 \over 2} k x_2^2}} = {{2 I \omega^2} \over {I \omega^2}}$
   • The terms ${1 \over 2}k$ on the left side and $I\omega^2$ on the right side cancel out: ${{x_1^2} \over {x_2^2}} = 2$
   • Taking the square root of both sides gives the ratio of the compression distances: ${{x_1} \over {x_2}} = \sqrt{2}$

6. Conclusion The ratio $x_1 / x_2$ is $\sqrt{2}$. This corresponds to option C.