a
(a) In figure $(b)$ current through ${R_2} = i - \frac{i}{{10}} = \frac{{9i}}{{10}}$
Potential difference across ${R_2}$ = Potential difference across $R$
$==>$ ${R_2} \times \frac{9}{{10}}i = R \times \frac{i}{{10}}$ i.e. ${R_2} = \frac{R}{9} = \frac{{11}}{9}\,\Omega $
${R_{eq}} = \frac{{{R_2} \times R}}{{({R_2} + R)}} = \frac{{\frac{{11}}{9} \times \frac{{11}}{1}}}{{\frac{{11}}{9} + \frac{{11}}{1}}} = \frac{{11}}{{10}}\,\Omega $
Total circuit resistance $ = \frac{{11}}{{10}} + {R_1} = R = 11$ $==>$ ${R_1} = 9.9\,\Omega $