If m : n is the duplicate ratio of m + x : n + x; show that : x² = mn.

According to question,

$\Rightarrow \dfrac{m}{n} = \dfrac{(m + x)^2}{(n + x)^2} \\[1em] \Rightarrow \dfrac{m}{n} = \dfrac{m^2 + x^2 + 2mx}{n^2 + x^2 + 2nx} \\[1em] \Rightarrow m(n^2 + x^2 + 2nx) = n(m^2 + x^2 + 2mx) \\[1em] \Rightarrow mn^2 + mx^2 + 2mnx = nm^2 + nx^2 + 2mnx \\[1em] \Rightarrow mx^2 - nx^2 = nm^2 - mn^2 + 2mnx - 2mnx \\[1em] \Rightarrow x^2(m - n) = mn(m - n) \\[1em] \Rightarrow x^2 = mn.$

Hence, proved that x² = mn.