A) 1 : 4
B) 1 : 2
C) 2 : 3
D) 3 : 4
Correct Answer: A
Solution :
\[G=\sqrt{ab}\] \[M=\frac{\frac{1}{a}+\frac{1}{b}}{2}\] \[M=\frac{a+b}{2ab}\] Given that\[\frac{1}{M}:G=4:5\] \[\frac{2ab}{(a+b)\sqrt{ab}}=\frac{4}{5}\] \[\Rightarrow \]\[\frac{a+b}{2\sqrt{ab}}=\frac{5}{4}\]\[\Rightarrow \]\[\frac{a+b+2\sqrt{ab}}{a+b-2\sqrt{ab}}=\frac{5+4}{5-4}\] {Using Componendo & Dividendo} \[\Rightarrow \]\[\frac{({{\sqrt{a)}}^{2}}+{{(\sqrt{b})}^{2}}+2\sqrt{ab}}{{{(\sqrt{a})}^{2}}+{{(\sqrt{b})}^{2}}-2\sqrt{ab}}=\frac{9}{1}\] \[\Rightarrow \]\[{{\left( \frac{\sqrt{b}+\sqrt{a}}{\sqrt{b}-\sqrt{a}} \right)}^{2}}=\frac{9}{1}\Rightarrow \frac{\sqrt{b}+\sqrt{a}}{\sqrt{b}-\sqrt{a}}=\frac{3}{1}\] \[\Rightarrow \]\[\Rightarrow \frac{\sqrt{b}+\sqrt{a}+\sqrt{b}-\sqrt{a}}{\sqrt{b}-\sqrt{a}-\sqrt{b}+\sqrt{a}}=\frac{3+1}{3-1}\] {Using Componendo & Dividendo} \[\sqrt{\frac{b}{a}}=\frac{4}{2}=2\] \[\frac{b}{a}=\frac{4}{1}\] \[\frac{a}{b}=\frac{1}{4}\Rightarrow a:b=1:4\]
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