\(=\frac{\mathrm{R}}{\sqrt{\mathrm{R}^{2}+\mathrm{X}_{\mathrm{L}}^{2}}}=\frac{\mathrm{R}}{\sqrt{\mathrm{R}^{2}+(2 \mathrm{R})^{2}}}=\frac{\mathrm{R}}{\sqrt{5 \mathrm{R}}}\)

Power factor \(_{(\text {new })}\)

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

\(=\frac{\mathrm{R}}{\sqrt{2} \mathrm{R}}\)

\(\therefore\) \(\frac{{New\,power\,factor}}{{Old\,power\,factor}} = \frac{{\frac{{\text{R}}}{{\sqrt {{\text{2R}}} }}}}{{\frac{{\text{R}}}{{\sqrt {5{\text{R}}} }}}} = \sqrt {\frac{5}{2}} \)