If the ratio of H.M. and G.M. between two num | vedclass

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(d) We have $H.M.$ = $\frac{{2ab}}{{a + b}}$ and $G.M.$ $ = \sqrt {ab} $

So $\frac{{H.M.}}{{G.M.}} = \frac{4}{5}$

$ \Rightarrow $$\frac{{2ab/(a

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$(b)$ and $(c)$ both

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(d) We have $H.M.$ = $\frac{{2ab}}{{a + b}}$ and $G.M.$ $ = \sqrt {ab} $

So $\frac{{H.M.}}{{G.M.}} = \frac{4}{5}$

$ \Rightarrow $$\frac{{2ab/(a + b)}}{{\sqrt {ab} }} = \frac{4}{5}$

$ \Rightarrow $ $\frac{{2\sqrt {ab} }}{{(a + b)}} = \frac{4}{5}$

$ \Rightarrow $$\frac{{a + b}}{{2\sqrt {ab} }} = \frac{5}{4}$

$ \Rightarrow $ $\frac{{a + b + 2\sqrt {ab} }}{{a + b - 2\sqrt {ab} }} = \frac{{5 + 4}}{{5 - 4}}$

$ \Rightarrow $$\frac{{{{(\sqrt a + \sqrt b )}^2}}}{{{{(\sqrt a - \sqrt b )}^2}}} = \frac{9}{1}$

$ \Rightarrow $ $\frac{{\sqrt a + \sqrt b }}{{\sqrt a - \sqrt b }} = \frac{3}{1}$

$ \Rightarrow $$\frac{{(\sqrt a + \sqrt b ) + (\sqrt a - \sqrt b )}}{{(\sqrt a + \sqrt b ) - (\sqrt a - \sqrt b )}} = \frac{{3 + 1}}{{3 - 1}}$

$ \Rightarrow $ $\frac{{2\sqrt a }}{{2\sqrt b }} = \frac{4}{2}$

$ \Rightarrow $$\left( {\frac{a}{b}} \right) = {2^2} = 4$

$ \Rightarrow $ $a:b = 4:1$ or $b:a = 1:4$.

Aliter : Let the numbers be in the ratio $\lambda :1$ and let they be $\lambda a$ and $a$

Then $\frac{{2(\lambda a)a}}{{\lambda a + a}}.\frac{1}{{\sqrt {\lambda a\;.\;a} }} = \frac{4}{5}$

$ \Rightarrow $$\frac{{\sqrt \lambda }}{{\lambda + 1}} = \frac{2}{5}$

$ \Rightarrow $ $25\lambda = 4({\lambda ^2} + 2\lambda + 1)$

$ \Rightarrow $$(\lambda - 4)(4\lambda - 1) = 0$

$ \Rightarrow $ $\lambda = 4$ or $\lambda = \frac{1}{4}$.

If the ratio of $H.M.$ and $G.M.$ between two numbers $a$ and $b$ is $4:5$, then the ratio of the two numbers will be

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