If both the A.M. between m and n and G.M. between two distinct positive numbers a and b are equal to ma+nbm+n, then n is equal to

$\frac{m a + n b}{m + n} = \frac{1}{2} (m + n) = \sqrt{a b}$ (1)

$\Rightarrow m a + n b = \sqrt{a b} (m + n)$

$\Rightarrow m \sqrt{a} (\sqrt{a} - \sqrt{b}) = n \sqrt{b} (\sqrt{a} - \sqrt{b})$

$\Rightarrow \frac{m}{\sqrt{b}} = \frac{n}{\sqrt{a}} = k (say)$ (2)

From (1) and (2)

$\frac{1}{2} k (\sqrt{a} + \sqrt{b}) = \sqrt{a b} \Rightarrow k = \frac{2 \sqrt{a b}}{\sqrt{a} + \sqrt{b}}$

Thus, $n = k \sqrt{a} = \frac{2 a \sqrt{b}}{\sqrt{a} + \sqrt{b}}$

If both the A.M. between m and n and G.M. between two distinct positive numbers a and b are equal to $\frac{m a + n b}{m + n}$ , then n is equal to