Answer
$x=6.92\;km$
Work Step by Step
The resistance of the wires from west end to short point is: $R_w=13x\;\Omega$
The resistance of the wires from east end to short point is: $R_e=13(10-x)\;\Omega$
Let $R$ be the resistance of the short
When the resistance of the wires and the short is measured from the east end, the equivalent resistance becomes $100\;\Omega$. Therefore,
$R_{eq}=2\times13(10-x)+R$
or, $2\times13(10-x)+R=100$
or, $260-26x+R=100$
or, $26x-R=160\;........................(1)$
When the resistance of the wires and the short is measured from the west end, the equivalent resistance becomes $200\;\Omega$. Therefore,
$R_{eq}=2\times13x+R$
or, $2\times13x+R=200$
or, $26x+R=200\;........................(2)$
Adding Eq. $1$ and Eq. $2$, we obtain
$52x=360$
or, $\boxed{x=6.92\;km}$
