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

power factor $\left(\mathrm{cos\varphi }\right)$ is a ratio of resistance and impedance of AC circuit.

Power factor of AC circuit is given by

$\mathrm{cos\varphi }=\frac{\mathrm{R}}{\mathrm{Z}}$                           …(i)

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

Equations (i) and (ii) give,

$\mathrm{cos\varphi }=\frac{\mathrm{R}}{\sqrt{{\mathrm{R}}^2+{\left({\mathrm{X}}_{\mathrm{L}}-{\mathrm{X}}_{\mathrm{C}}\right)}^2}}$   …(iii)

Given, $\mathrm{R}=8 \mathrm{\Omega }, {\mathrm{X}}_{\mathrm{L}}=31 \mathrm{\Omega }, {\mathrm{X}}_{\mathrm{C}}=25 \mathrm{\Omega }$

$\mathrm{cos\varphi }=\frac{8}{\sqrt{(8{)}^2+(31-25{)}^2}}=\frac{8}{\sqrt{64+36}}$

$\mathrm{cos\varphi }=0.80$