Question

If a, b, c are positive numbers such that a>b>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

The correct option is A

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

$Letf\left(x\right)=(a+b-2c)x^2+(b+c-2a)x+(c+a-2b)$

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 $\Rightarrow$ a - b > 0, a - c >0]

i.e b > $\frac{a+c}{2}$

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