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Let alpha and beta be the roots of equa...

Let `alpha and beta ` be the roots of equation `px^(2) + qx + r = 0, p ne 0`. If p, q , r are in A.P. and `(1)/(alpha) + (1)/(beta)` = 4, then the value of `| alpha - beta| ` is :

A

`(sqrt(34))/9`

B

`(2sqrt(13))/9`

C

`(sqrt(61))/9`

D

`(2sqrt(17))/9`

Text Solution

Verified by Experts

The correct Answer is:
B, D
`:'1/(alpha)+1/(beta)=4implies(alpha+beta)/(alpha beta)=4`
`implies(-q/p)/(r/p)=4`
`impliesq=-4r`………….i
Also given `p,q,r` are in AP.
`:.2q=p+r`
`impliesp=-9r` [from Eq (i) ].ii
Now `|alpha-beta|=(sqrt(D))/(|a|)[:' "for" ax^(2)+bx+c=0, alpha-beta=(sqrt(D))/a]`
`=(sqrt((q^(2)-4pr)))/(|p|)`
`=(sqrt((16r^(2)+36r^(2)))/(9|r|)=(sqrt(52)|r|)/(9|r|)`[from Eqs i and ii]
`=(2sqrt(13))/9`
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