We have,

Case I : `R=12Omegal_(2)=120cm`

Case II : `R=18Omega,l_(2)=150cm`

From first condition,

Internal resistance (r) `=R((l_(1)-l_(2))/(l_(2)))`

`r=12((l_(1)-l_(2))/(120))" "......(i)`

From second condition,

Internal resistance (r) `=R((l_(1)-l_(2))/(l_(2)))`

`=18((l_(1)-150)/(150))" "......(ii)`

From equations (i) and (ii)

`12((l_(1)-120)/(120))=18((l_(1)-150)/(150))`

`5(l_(1)-120)=6(l_(1)-150)`

`5l_(1)-600=6l_(1)-900`

`l_(1)=300cm.`

Putting the value of `l_(2)` in equation (i) we, get

`r=12((300-120)/(120))`

`r=18Omega`

Therefore, the balancing lenght is 300 cm and internal resistance is `18Omega`.