The resistance of the upper portion is

\(R_{1}=x l_{1}\)

The resistance of the lower portion is

\(R_{2}=x l_{2}\)

Equivalent resistance between \(A\) and \(B\) is

\(R = \frac{{{R_1}{R_2}}}{{{R_1} + {R_2}}} = \frac{{\left( {x{l_1}} \right)(x{l_2})}}{{x{l_1} + x{l_2}}}\)

\(\frac{8}{3}=\frac{x l_{1} l_{2}}{l_{1}+l_{2}}\) or \(\frac{8}{3}=\frac{x l_{1} l_{2}}{l_{2}\left(\frac{l_{1}}{l_{2}}+1\right)}\) or \(\frac{8}{3}=\frac{x_{1}}{\left(\frac{l_{1}}{l_{2}}+1\right)}\)        ......\((i)\)

Also \(\quad R_{0}=x l_{1}+x l_{2}\)

\({12=x\left(l_{1}+l_{2}\right)}\)

\({12=x l_{2}\left(\frac{l_{1}}{l_{2}}+1\right)}\)        ....\((ii)\)

Divide \((i)\) by \((ii),\) we get

\(\frac{{\frac{8}{3}}}{{12}} = \frac{{\frac{{x{l_1}}}{{\left( {\frac{{{l_1}}}{{{l_2}}} + 1} \right)}}}}{{x{l_2}\left( {\frac{{{l_1}}}{{{l_2}}} + 1} \right)}}{\rm{ or }}\frac{8}{{36}} = \frac{{{l_1}}}{{{l_2}{{\left( {\frac{{{l_1}}}{{{l_2}}} + 1} \right)}^2}}}\)

\(\left(\frac{l_{1}}{l_{2}}+1\right)^{2} \frac{8}{36}=\frac{l_{1}}{l_{2}}\) or \(\left(\frac{l_{1}}{l_{2}}+1\right)^{2} \frac{2}{9}=\frac{l_{1}}{l_{2}}\)

Let \(y=\frac{l_{1}}{l_{2}}\)

\(\therefore\)       \(2(y+1)^{2}=9 y\) or \(2 y^{2}+2+4 y=9 y\)

or    \( 2 y^{2}-5 y+2=0\)

Solving this quadratic equation, we get

\(y=\frac{1}{2}\)  or  \(2 \therefore \frac{l_{1}}{l_{2}}=\frac{1}{2}\)