In a series LR circuit X _ L = R and power factor of the circuit is...

In case of $L - R$ circuit

$Z = \sqrt{{{X}_{L}}^{2} + {R}^{2}}$ Therefore, power factor

${P}_{1} = cosϕ = \dfrac{R}{Z}$

As ${X}_{L} = R$ $⇒ Z = \sqrt{2}R$

$⇒ {P}_{1} = \dfrac{R}{\sqrt{2}R} ⇒ {P}_{1} = \dfrac{1}{\sqrt{2}}$

In case of $L - C - R$ circuit

$Z = \sqrt{{R}^{2} + {\left({X}_{L} - {X}_{C}\right)}^{2}}$

According to the question ${X}_{L} = {X}_{C}$ $⇒ Z = R$ $⇒ {P}_{2} = cosϕ = \dfrac{R}{R} = 1$

$⇒ \dfrac{{P}_{1}}{{P}_{2}} = \dfrac{1}{\sqrt{2}}$