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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|` is :

A

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

B

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

C

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

D

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

Text Solution

Verified by Experts

The correct Answer is:
4
We have `(1)/(alpha ) + (1)/(beta) = 4 and 2 q + r `
`therefore (2q)/(p) = 1 + (r)/(p)`
`rArr - 2 (alpha + beta) = 1 + alpha beta`
`rArr - 2 ((1)/(alpha ) + (1)/(beta)) = (1)/(alpha beta) + 1`
`rArr (1)/(alpha beta) = - 9 `
` therefore alpha beta = - (1)/(9)`
` therefpre alpha + beta = - (4)/(9)` ,brgt Equation having roots ` alpha , beta is 9x^(2) + 4x - 1 = 0 `
` therefore |alpha - beta| = sqrt((alpha + beta)^(2) - 4alpha beta)`
`= sqrt((16)/(81)+ (4)/(9))`
` (2sqrt(13))/(9)`
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