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

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|`

A

`(sqrt(34))/(9)`

B

`(2sqrt(13))/(9)`

C

`(sqrt(61))/(9)`

D

`(2sqrt(17))/(9)`

Text Solution

Verified by Experts

Since `alpha and beta` are roots of the the equation `px^(2)+qx+r=0`. Therefore, `alpha+beta = - (q)/(p) and alpha beta = (r)/(p)`
Now, `(1)/(alpha)+(1)/(beta)=4 rArr (alpha+beta)/(alpha beta)=4 rArr -(q)/(r)=4 rArr q =-4r`
It is given that p,q, r are in AP. Therefore, `2q = p+r rArr -8r = p+r rArr p = -9r`
`therefore" "alpha + beta = -(q)/(p) = -(4)/(9) and alpha beta =(r)/(p) = -(1)/(9)`
Now, `(alpha - beta)^(2) = (alpha+beta)^(2) - 4 alpha beta`
`rArr" "(alpha-beta)^(2)=(16)/(81)+(4)/(9)=(52)/(81) rArr |alpha-beta|=(2 sqrt(13))/(9)`
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