If $m : n$ is the duplicate ratio of $(m + x)^{2} : (n + x)^{2}$ show that $x^{2} = m n$ .

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Solution

Given that,
$m : n = (m + x)^{2} : (n + x)^{2}$
$\frac{m}{n} = \frac{(m + x)^{2}}{[n + x]^{2}}$
$m [n + x]^{2} = [m + x]^{2} \times n$
$m n^{2} + m x^{2} + 2 m n x = m^{2} n + x^{2} n + 2 m n x$
$x^{2} [m - n] = m n (m - n)$
$x^{2} = m n .$

Hence proved.

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