1. Initial series $LR$ circuit

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

   Given: $X_L = R$

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

   The 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 in series

   Now the circuit becomes a series $LRC$ circuit.

   Given: $X_L = X_C$

   Hence the net reactance is $X = X_L - X_C = 0$

   Therefore the impedance becomes purely resistive: $Z_2 = R$

   So the new power factor is $P_2 = \cos\phi_2 = \frac{R}{Z_2} = \frac{R}{R} = 1$

3. Required ratio

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

4. Option check

   • A: $\frac{1}{2}$ ✗
   • B: $\frac{1}{\sqrt{2}}$ ✓
   • C: $\frac{\sqrt{3}}{\sqrt{2}}$ ✗
   • D: $2:1$ ✗

Therefore, the correct answer is Option B.