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Let a ,b ,c be real numbers a!=0. If alp...

Let `a ,b ,c` be real numbers `a!=0.` If `alpha` is a root of `a^2x^2+b x+c=0.beta` is the root of `a^2x^2-b x-c=0a n d0

A

`gamma=(alpha+beta)/(2)`

B

`gamma=alpha+(beta)/(2)`

C

`gamma=alpha`

D

`alphaltgammalt beta`

Text Solution

Verified by Experts

The correct Answer is:
D
Since, `alpha` is a root of `a^(2)x^(2)+bx+c=0`
`impliesa^(1)alpha^(2)+balpha+c=0" "...(i)`
`and beta"is a root of"a^(2)x^(2)-bx-c=0`
`impliesa^(2)beta^(2)-b beta-c=0" "...(ii)`
Let `f(x)=a^(2)alpha^(2)+2bx+2c`
`therefore f(alpha)a^(2)alpha^(2)+2balpha+2c`
`=a^(2)alpha^(2)-2a^(2)alpha^(2)=-a^(2)alpha^(2)" "["from Eq."(i)]`
and `f(beta)=alpha^(2)beta^(2)+2b beta+2c`
`= a^(2)beta^(2)+2a^(2)beta^(2)=3a^(2)beta^(2)" "["from Eq."(ii).]`
`implies f(alpha) f(beta)lt0`
f(x) must have a root laying in the open interval `(alpha, beta).`
`therefore alpha lt gamma lt beta`
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