Multiplying ${1^{st}}$ equation by a and ${2^{nd}}$ equation by $\mathrm{r}$ and then subtracting we get;

$\frac{\mathrm{rV}_{2}-\mathrm{aV}_{1}}{\mathrm{K}}=\mathrm{q}_{1}\left(\frac{\mathrm{r}}{\mathrm{a}}-\frac{\mathrm{a}}{\mathrm{r}}\right)=\mathrm{q}_{1} \frac{\mathrm{r}^{2}-\mathrm{a}^{2}}{\mathrm{ar}}$

$\therefore {{\rm{q}}_1} = \frac{{\left( {{\rm{r}}{{\rm{V}}_2} - {\rm{a}}{{\rm{V}}_1}} \right){\rm{ra}}}}{{\left( {{{\rm{r}}^2} - {{\rm{a}}^2}} \right){\rm{K}}}}$

Similarly, we find, $q_{2}=\frac{\left(r V_{1}-a V_{2}\right) r a}{\left(r^{2}-a^{2}\right) K}$