Let `alpha,beta` be the roots of the equation `x^(2)-px+r=0 and alpha//2,2beta` be the roots of the equation `x^(2)-qx+r=0`, then the value of r is (1)`(2)/(9)(p-q)(2q-p)` (2) `(2)/(9)(q-p)(2p-q)` (3)`(2)/(9)(q-2p)(2q-p)` (4)`(2)/(9)(2p-q)(2q-p)`

Options

A. `(2)/(9)(p-q)(2q-p)`
B. `(2)/(9)(q-p)(2p-q)`
C. `(2)/(9)(q-2p)(2q-p)`
D. `(2)/(9)(2p-q)(2q-p)`

Answer

D

Explanation

Since `alpha, beta` are the roots of `x^(2) - px + r = 0`
`therefore" "(alpha)/(2)+2beta = q and (alpha)/(2) xx 2 beta r`
Solving `alpha + beta = p and (alpha)/(2)+2 beta = q`, we get
`alpha =(2)/(3)(2p-q) and beta = (1)/(3)(2q-p)`
`therefore" "alpha beta = r rArr r = (2)/(9) (2p-q) (2q-p)`