Sequence And Series Question 271
Question: 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
[IIT 1992; MP PET 2000]
Options:
A) $ 1:2 $
B) $ 2:1 $
C) $ 4:1 $
D) $ 1:4 $
Show Answer
Answer:
Correct Answer: D
Solution:
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 $ $ ( \frac{a}{b} )=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} $ .
BETA
