Let x be resistance per unit length of the wire. Then, the resistance of the upper portion is, R1=xl1 the resistance of the lower portion is, R2=xl2 Equivalent resistance between A and B is R=
R1R2
R1+R2
=
(xl1)(xl2)
xl1+xl2
8
3
=
xl1l2
l1+l2
or
8
3
=
xl1l2
l2(
l1
l2
+1)
or
8
3
=
xl1
(
l1
l2
+1)
......(i) (i)Also R0=xl1+xl2 12=x(l1+l2) 12=xl2(
l1
l2
+1) .....(ii) Divide (i) by (ii), we get
8
3
12
=
xl1
(
l1
l2
+1)
xl2(
l1
l2
+1)
or
8
36
=
l1
l2(
l1
l2
+1)2
(
l1
l2
+1)2
8
36
=
l1
l2
or (
l1
l2
+1)2
2
9
=
l1
l2
Let y=
l1
l2
∴2(y+1)2=9y or 2y2+2+4y=9y or 2y2−5y+2=0 Solving this quadratic equation, we get y=