Explanation:
D Given,
Two positive numbers a \& \(\mathrm{b}\) in ratio of \(4: 5\)
We know that,
\(\text { H.M. }=\frac{2 a b}{a+b}, \text { G.M. }=\sqrt{a b},\)
Now, \(\frac{\text { H.M. }}{\text { G.M. }}=\frac{\frac{2 \mathrm{ab}}{\mathrm{a}+\mathrm{b}}}{\sqrt{\mathrm{ab}}}=\frac{4}{5}\)
\(\frac{2 \sqrt{\mathrm{ab}}}{\mathrm{a}+\mathrm{b}}=\frac{4}{5}\)
\(5 \sqrt{\mathrm{ab}}=2(\mathrm{a}+\mathrm{b})\)
On squaring both side,
\(25 a b=4(a+b)^2\)
\(4 a^2-17 a b+4 b^2=0\)
\((4 a-b)(a-4 b)=0\)
\(4 a=b\)
\(\frac{a}{b}=\frac{1}{4}\)
\(\therefore \quad a: b=1: 4\)