In a series L R circuit X L = R and power factor of the circuit is P 1 . When capacitor with capacitance C…

In case of $\mathrm{L}-\mathrm{R}$ circuit

$Z=\sqrt{{X_L}^2+R^2}$ Therefore, power factor

$P_1=\cos \varphi =\frac{R}{Z}$

As $X_L=R$ $\Rightarrow Z=\sqrt{2}R$

$\Rightarrow P_1=\frac{R}{\sqrt{2}R}\Rightarrow P_1=\frac{1}{\sqrt{2}}$

In case of  $\mathrm{L}-\mathrm{C}-\mathrm{R}$ circuit

$Z=\sqrt{R^2+{\left(X_L-X_C\right)}^2}$

According to the question $X_L=X_C$ $\Rightarrow Z=R$  $\Rightarrow P_2=\cos \varphi =\frac{R}{R}=1$

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