If a, b, c are positive numbers such tha...

If `a`, `b`, `c` are positive numbers such that `a gt b gt c` and the equation `(a+b-2c)x^(2)+(b+c-2a)x+(c+a-2b)=0` has a root in the interval `(-1,0)`, then

A

`b` cannot be the `G.M.` of `a`,`c`

B

`b` may be the `G.M.` of `a`,`c`

C

`b` is the `G.M.` of `a`,`c`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:

A

`(a)` Let `f(x)=(a+b-2c)x^(2)+(b+c-2a)x+(c+a-2b)`
According to the given condition, we have
`f(0)f(-1) lt 0`
`implies (c+a-2b)(2a-b-c) lt 0`
`implies (c+a-2b)(a-b+a-c) lt 0`
`implies c+a-2b lt 0`
[`a gt b gt c`, given `implies a-b gt 0`, `a-c gt 0`]
` b gt (a+c)/(2)`
`impliesb` cannot be the `G.M.` of `a`, `c`, since `G.M lt A.M.` always.

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