1. Resonant frequency of an LC oscillator

   The angular resonant frequency is

   $$\omega = \frac{1}{\sqrt{LC}}.$$

   Initially,

   $$\omega_0 = \frac{1}{\sqrt{LC}}.$$

2. New values of inductance and capacitance

   Given:

   $$L' = 2L, \qquad C' = 8C.$$

3. New resonant frequency

   $$\omega' = \frac{1}{\sqrt{L'C'}} = \frac{1}{\sqrt{(2L)(8C)}}.$$ $$\omega' = \frac{1}{\sqrt{16LC}} = \frac{1}{4\sqrt{LC}}.$$

   Since

   $$\omega_0 = \frac{1}{\sqrt{LC}},$$

   therefore,

   $$\omega' = \frac{1}{4}\,\omega_0.$$

4. Find $x$

   We are given

   $$\omega' = x\,\omega_0.$$

   Comparing,

   $$x = \frac{1}{4}.$$

5. Option check

   • A: $\frac{1}{4}$ ✅
   • B: $\frac{1}{16}$ ❌
   • C: $4$ ❌
   • D: $16$ ❌

Final Answer: $x = \frac{1}{4}$