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To see why the relation has to hold, you have to acknowledge that it needs to transform the line elements you have written down into each other. One can see that this is true by taking the logarithm on both sides, which yields $$\rho=\log{x^i}-\log{z}.$$ Taking the derivative of this expression and squaring it gives $\mathrm{d}\rho^2=\mathrm{d}z^2/z^2$, ...