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If (1)/(a)+(1)/(c )=(1)/(2b-a)+(1)/(2b-c...

If `(1)/(a)+(1)/(c )=(1)/(2b-a)+(1)/(2b-c)`, then

A

`a,b,c` are in `A.P.`

B

`a,(b)/(2),c` are in `A.P.`

C

`a,(b)/(2),c` are in `H.P.`

D

`a,2b,c` are in `H.P.`

Text Solution

Verified by Experts

The correct Answer is:
A, D
`(a,d)` `(1)/(a)+(1)/(c )=(1)/(2b-a)+(1)/(2b-c)`
`implies(1)/(a)+(1)/(a-2b)+(1)/(c )+(1)/(c-2b)=0`
`implies((1)/(a)+(1)/(c-2b))+((1)/(a-2b)+(1)/(c ))=0`
`implies(a+c-2b)((1)/(a(c-2b))+(1)/(c(a-2b)))=0`
Either, `a+c-2b=0`
`impliesa,b,c` are in `A.P.` or `(1)/(a(c-2b))+(1)/(c(a-2b))=0`
`impliesac-2bc+ac-2ab=0`
`implies2ac=2b(a+c)`
`impliesb=(ac)/(a+c)impliesa,2b,c` are in `H.P`.
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