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Q.

Let G be the geometric mean of two positive numbers a and b, and M be the arithmetic mean of $\frac{1}{a}$ and $\frac{1}{b} .$ If $\frac{1}{M} : G$ is 4 : 5, then a : b can be

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a

1 : 4

b

3 : 4

c

2 : 3

d

1 : 2

answer is A.

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Detailed Solution

$G = \sqrt{ab} M = \frac{\frac{1}{a} + \frac{1}{b}}{2} = \frac{b + a}{2 ab} \frac{1}{M} : G = 4 : 5 \Rightarrow \frac{2 ab}{(a + b) \sqrt{ab}} = \frac{4}{5} \Rightarrow \frac{\sqrt{ab}}{a + b} = \frac{2}{5} 25 ab = 4 a^{2} + 4 b^{2} + 8 ab \Rightarrow 4 a^{2} - 17 ab + 4 b^{2} = 0 \Rightarrow (4 a - b) (a - 4 b) = 0 \Rightarrow 4 a = b or a = 4 b$

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