1. Initial series LR circuit

   Given: $X_L = R$

   For a series $LR$ circuit, impedance is $Z_1 = \sqrt{R^2 + X_L^2}$

   Since $X_L = R$, $Z_1 = \sqrt{R^2 + R^2} = R\sqrt{2}$

   Power factor is $P_1 = \cos\phi_1 = \frac{R}{Z_1} = \frac{R}{R\sqrt{2}} = \frac{1}{\sqrt{2}}$

2. After adding capacitor

   Now the circuit becomes a series $LRC$ circuit.

   Given: $X_C = X_L$

   Net reactance: $X = X_L - X_C = 0$

   Therefore impedance becomes purely resistive: $Z_2 = R$

   Hence power factor: $P_2 = \cos\phi_2 = \frac{R}{Z_2} = \frac{R}{R} = 1$

3. Ratio $P_1 : P_2$

   $P_1 : P_2 = \frac{1}{\sqrt{2}} : 1 = 1 : \sqrt{2}$

4. Option check

   • A: $1 : \sqrt{2}$ ✅
   • B: $1 : 3$ ❌
   • C: $1 : 2$ ❌
   • D: $1 : 1$ ❌

Therefore, the correct answer is A.