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If A1,A2,G1,G2 and H1 ,H2 be two AMs,GMs...

If `A_1,A_2,G_1,G_2` and `H_1 ,H_2` be two `AMs,GMs` and `HMs` between two quantities then the value of `(G_1G_2)/(H_1H_2)` is

A

` (A_(1)+A_(2))/(H_(1)+H_(2))`

B

` (A_(1)-A_(2))/(H_(1)+H_(2))`

C

`(A_(1)+A_(2))/(H_(1)-H_(2))`

D

` (A_(1)-A_(2))/(H_(1)-H_(2))`

Text Solution

Verified by Experts

The correct Answer is:
a
Let the two quantities be a and b . Then , `a_(1),A_(1),A_(2),b` are in AP .
` :. A_(1) -a = b -A_(2) rArr A_(1) +A_(2) = a+b " " ` …(i)
Again `a_(1),H_(1),H_(2),b` are in HP
` :. (G_(1))/a = b/(G_(2)) rArr G_(1) G_(2) = ab " "` ...(ii)
Also ` a,H_(1),H_(2),b ` are in HP.
` :. 1/(H_(1)) - 1/a = 1/b -1/(H^(2)) rArr 1/(H_(1)) +1/(H_(2)) = 1/b +1/a`
` rArr (H_(1) +H_(2))/(H_(1)H_(2))=(a+b)/(Ab) rArr (H_(1)+H_(2))/(H_(1)H_(2)) = (A_(1)+A_(2))/(G_(1)G_(2))`
[from Eqs.(i) and (ii)]
` rArr (G_(1)G_(2))/(H_(1)H_(2)) = (A_(1)+A_(2))/(H_(1) H_(2))`
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