Question

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

Answer 1

Correct Answer - 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.