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Question

If a, b, c are positive numbers such that a>b>c and the equation (a+b2c)x2+(b+c2a)x+(c+a2b)=0 has a root in the interval (-1, 0), then


A

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

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B

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

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C

b is the G.M. of a, c

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D

none of these

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Solution

The correct option is A

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


Let f(x)=(a+b2c)x2+(b+c2a)x+(c+a2b)

According to the given condition, we have

f (0) f (-1) < 0

i.e. (c +a -2b) (2a - b - c) < 0

i.e. (c +a -2b) (a - b + a - c) < 0

i.e. c + a - 2b < 0 [a > b > c, given a - b > 0, a - c >0]

i.e b > a+c2

b cannot be the G.M. of a, c, since G.M < A.M. always.


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