Question

[Key Concept: Series RC impedance is √(R² + X_C²); for parallel RC, 1/Z = √(1/R² + 1/X_C²).]

in series

$$\begin{array}{r} Z_{s} = \sqrt{R^{2} + \left( \frac{1}{\omega C} \right)^{2}}\#(i) \end{array}$$

In parallel

$$\begin{array}{r} \frac{1}{Z_{p}} = \sqrt{\frac{1}{R^{2}} + (\omega C)^{2}}\#(i) \end{array}$$

here $Z_{S} = 2Z_{P}$

$$\begin{matrix} & R^{2} + \frac{1}{4\pi f^{2}C^{2}} = 4\frac{1}{\left( \frac{1}{R^{2}} + 4\pi^{2}f^{2}C^{2} \right)} \\ & \left( R^{2} + \frac{1}{4\pi^{2}f^{2}C^{2}} \right)\left( \frac{1}{R^{2}} + 4\pi^{2}f^{2}C^{2} \right) = 4 \\ & 1 + 1 + 4\pi f^{2}R^{2}C^{2} + \frac{1}{4\pi^{2}f^{2}C^{2}R^{2}} = 4 \\ & 4\pi f^{2}R^{2}C^{2} + \frac{1}{4\pi^{2}f^{2}C^{2}R^{2}} = 2 \end{matrix}$$

On solving

$$\begin{matrix} & 4\pi^{2}f^{2}R^{2}C^{2} = 1 \\ & f = \frac{1}{2\pi RC} \end{matrix}$$