A ring is made of  a wire having a resistance R0=12 Ω. Find the points A and B as shown in fig. at which a current carrying conductor should be connected so that the resistance ‘R’ of the sub circuit between these points is equal to   83 Ω

Let say Resistance of length l1 is R1 and Resistance of length l2 is R2 If ring is open, then resistors are in series R1+R2=R0=12-----(1)  If ring is closed, then resistors are in parallel between A and B R1R2R1+R2=83⇒R1R212=83  ⇒R1R2=32  Also,  R1−R22=R1+R22−4R1R2=122−4×32 = 144-128 = 16  ⇒R2−R1=4----(2) From (1) and (2) we get  Hence, R2=8 Ω , R1=4 Ω ∴ R1R2=l1l2=48=12