In LCR circuit, at a particular angular frequency the capacitive reactance and inductive reactance is same. If the angular frequency is doubled, the ratio of the reactance of the capacitor to that of the inductor will be

Let the initial angular frequency be $\omega_0$.

At $\omega_0$, the capacitive reactance $X_C$ and inductive reactance $X_L$ are equal:

$X_C = X_L \Rightarrow \dfrac{1}{\omega_0 C} = \omega_0 L$

When the angular frequency is doubled, the new angular frequency is $\omega = 2\omega_0$.

The new capacitive reactance is:

$X_C' = \dfrac{1}{\omega C} = \dfrac{1}{2\omega_0 C} = \dfrac{X_C}{2}$

The new inductive reactance is:

$X_L' = \omega L = 2\omega_0 L = 2X_L$

The ratio of the new capacitive reactance to the new inductive reactance is:

$\dfrac{X_C'}{X_L'} = \dfrac{\dfrac{X_C}{2}}{2X_L} = \dfrac{1}{4} \left(\dfrac{X_C}{X_L}\right)$

Since $X_C = X_L$, the ratio becomes $\dfrac{1}{4}$.