Sequence And Series Question 240
Question: If the ratio of H.M. and G.M. of two quantities is $ 12:13 $ , then the ratio of the numbers is
[RPET 1990]
Options:
A) $ 1:2 $
B) $ 2:3 $
C) $\frac{3}{4}$
D) None of these
Show Answer
Answer:
Correct Answer: D
Solution:
Given that $ \frac{H.M.}{G.M.}=\frac{12}{13} $
$ \Rightarrow $ $ {{(p-r)}^{2}}={{(p+r)}^{2}}-4pr=4K^{2}-4(-2K^{2})=12K^{2} $ or $ \frac{a+b}{2\sqrt{ab}}=\frac{13}{12} $
$ \Rightarrow $ $ \frac{(a+b)+2\sqrt{ab}}{(a+b)-2\sqrt{ab}}=\frac{13+12}{13-12}=\frac{25}{1} $
$ \Rightarrow $ $ 2p=(-2\pm 2\sqrt{3})K $
$ \Rightarrow $ $ \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}-\sqrt{b}}=\frac{5}{1} $
$ \Rightarrow $ $ \frac{(\sqrt{a}+\sqrt{b})+(\sqrt{a}-\sqrt{b})}{(\sqrt{a}+\sqrt{b})-(\sqrt{a}-\sqrt{b})}=\frac{5+1}{5-1} $
$ \Rightarrow $ $ \frac{2\sqrt{a}}{2\sqrt{b}}=\frac{6}{4} $
$ \Rightarrow $ $ {{( \frac{a}{b} )}^{1/2}}=\frac{\sqrt{a}}{\sqrt{b}} $
$ \Rightarrow $ $ a:b=9:4 $
BETA
