A potentiometer \(PQ\) is set up to compare two resistances as shown in the figure. The ameter \(A\) in the circuit reads \(1.0\, A\) when two way key \(K_3\) is open. The balance point is at a length \(l_1\, cm\) from \(P\) when two way key \(K_3\) is plugged in between \(2\) and \(1\) , while the balance point is at a length \(l_2\, cm\) from \(P\) when key \(K_3\) is plugged in between \(3\) and \(1\) . The ratio of two resistances \(\frac{{{R_1}}}{{{R_2}}}\) is found to be

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(D) \(\frac{{{l_1}}}{{{l_2} - {l_1}}}\)

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When key is at point \(( 1)\)

\(\mathrm{V}_{1}=\mathrm{i} \mathrm{R}_{1}=\mathrm{x} l_{1}\)

When key is at \(( 3)\)

\({{\rm{v}}_2} = {\rm{i}}\left( {{{\rm{R}}_1} + {{\rm{R}}_2}} \right) = {\rm{x}}{l_2}\)

\(\frac{\mathrm{R}_{1}}{\mathrm{R}_{1}+\mathrm{R}_{2}}=\frac{l_{1}}{l_{2}} \Rightarrow \frac{\mathrm{R}_{1}}{\mathrm{R}_{2}}=\frac{l_{1}}{l_{2}-l_{1}}\)

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