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

Options

A. `(sqrt(34))/9`
B. `(2sqrt(13))/9`
C. `(sqrt(61))/9`
D. `(2sqrt(17))/9`

Answer

B, D

Explanation

`:'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`