A coil of inductive reactance $$31\,\Omega $$  has a resistance of $$8\,\Omega .$$  It is placed in series with a condenser of capacitative reactance $$25\,\Omega .$$ The combination is connected to an $$AC$$  source of $$110\,V.$$  The power factor of the circuit is

Solution :
Power factor of $$AC$$ circuit is given by
$$\cos \phi = \frac{R}{Z}\,\,......\left( {\text{i}} \right)$$
where, $$R$$ is resistance and $$Z$$ is the impedance of the circuit and is given by
$$Z = \sqrt {{R^2} + \left( {{X_L} - X_{_C}^2} \right)} \,......\left( {{\text{ii}}} \right)$$
$${{X_L}} =$$  inductive reactance
$${{X_C}} =$$  capacitive reactance
Eqs. (i) and (ii) meet to give,
$$\cos \phi = \frac{R}{{\sqrt {{R^2} + {{\left( {{X_L} - {X_C}} \right)}^2}} }}\,......\left( {{\text{iii}}} \right)$$
Given, $$R = 8\,\Omega ,{X_L} = 31\,\Omega ,{X_C} = 25\,\Omega $$
$$\eqalign{ & \therefore \cos \phi = \frac{8}{{\sqrt {{{\left( 8 \right)}^2} + {{\left( {31 - 25} \right)}^2}} }} = \frac{8}{{\sqrt {64 + 36} }} \cr & {\text{Hence,}}\,\,\cos \phi = 0.80 \cr} $$