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If H(1) and H(2) are two harmonic means ...

If `H_(1)` and `H_(2)` are two harmonic means between two positive numbers a and b `(a != b)` , A and G are the arithmetic and geometric menas between a and b , then `(H_(2)+H_(1))/(H_(2)H_(1))` is

A

`A/G`

B

`(2A)/G`

C

`A/(2 G^(2))`

D

`(2A)/(G^(2))`

Text Solution

Verified by Experts

The correct Answer is:
d
Since , a, `H_(1),H_(2),b ` are in HP
` :. H_(1) = (3ab)/(a+2b) , H_(2) = (3ab)/(2a+b)`
Now, ` (H_(1)+H_(2))/(H_(1)H_(2)) = 1/(H_(2)) + 1/(H_(1)) = (2a+b)/(3ab) + (a+2b)/(3ab) = (a+b)/(ab) " " ` …(i)
Also, ` 2A = a+b" "`...(ii)
and `ab = G^(2)" "`...(iii)
From Eqs.(i) ,(ii) and (iii),
` (H_(1)+H_(2))/(H_(1)H_(2))=(2A)/(G^(2))`
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