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. `(sqrt61)/(9)`
B. `(2sqrt17)/(9)`
C. `(sqrt34)/(9)`
D. `(2sqrt13)/(9)`

Answer

D

Explanation

If `ax^(2)+bx+c=0` has roots `alpha and beta,` then `alpha + beta=-b//aand alpha beta=c/a.` Find the values of `alpha + beta and alpha beta` and then put in `(alpha-beta)^(2)=(alpha+beta)^(2)-4alpha beta` to get required value.
Given `alpha and beta` are roots of `px^(2)+qx+r=0,p ne 0.`
`therefore alpha+beta=(-q)/(p)'alphabeta=r/p" "...(i)`
Since, p, q and r in AP.
`therefore 2q=p+r" "...(ii)`
Also, `(1)/(alpha)+(1)/(beta)=4implies(alpha+beta)/(alphabeta)=4`
`impliesalpha+beta=4 alphabetaimplies(-q)/(p)=(4r)/(p)" "["fromEq."(i)]`
`impliesq=-4t`
On putting the vaue of q in Eq. (ii) we get
`implies2(-4r)=p+rimpliesp=-9r`
Now, `alpha+beta=(-q)/(p)=(4r)/(p)=(4r)/(-9r)=-4/9`
and `alphabeta=r/b=(r)/(-9r)=(1)/(-9)`
`therefore(alpha-beta)^(2)=(alpha=beta)^(2)-4alphabeta=(16)/(81)+4/9=(16+36)/(81)`
`implies(alpha-beta)^(2)=52/81`
`implies|alpha-beta|=2/9sqrt13`