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Given that alpha,gamma are roots of the ...

Given that `alpha,gamma` are roots of the equation `A x^2-4x+1=0,a n dbeta,delta` the roots of the equation of `B x^2-6x+1=0,` such that `alpha,beta,gamma,a n ddelta` are in H.P., then a.`A=3` b. `A=4` `B=2` d. `B=8`

A

`A=5`

B

`A=3`

C

`B=8`

D

`B=-8`

Text Solution

Verified by Experts

The correct Answer is:
B
`(b,c)` `alpha,beta,gamma,delta` are in `H.P.`
`implies(1)/(alpha)`, `(1)/(beta)`, `(1)/(gamma)`, `(1)/(delta)` are in `A.P.`
Let `d` be the common difference of the `A.P.`
Since `alpha`, `gamma` are roots of `Ax^(2)-4x+1=0`
`:.(alpha+gamma)/(alphagamma)=(4//A)/(1//A)=4`
or `(1)/(alpha)+(1)/(gamma)=4`
`implies(1)/(alpha)+(1)/(alpha)+2d=4` or `(1)/(alpha)+d=2`.......`(i)`
Also `beta`, `delta` are roots of `Bx^(2)-6x+1=0`
`:.(beta+delta)/(betadelta)=(1)/(beta)+(1)/(delta)=(6//B)/(1//B)=6` or `(1)/(alpha)+d+(1)/(alpha)+3d=6`
`implies(1)/(alpha)+2d=3`..........`(ii)`
Solving `(i)` and `(ii)`, we get `(1)/(alpha)=1` and `d=1`
`:. (1)/(alpha)=1`, `(1)/(beta)=2`, `(1)/(gamma)=3` and `(1)/(delta)=4`
since `(1)/(alphagamma)=AimpliesA=3`
Also, `(betagamma)=BimpliesB=8`
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