Let `alpha,beta` be the roots of the equation `x^2-p x+r=0a n dalpha//2,2beta` be the roots of the equation `x^2-q x+r=0.` Then the value of `r` is `2/9(p-q)(2q-p)` b. `2/9(q-p)(2q-p)` c. `2/9(q-2p)(2q-p)` d. `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

The equation `x^(2)-px+r=0` has roots `(alpha, beta)` and the equation `x^(2)-qx+r` has roots `((alpha)/2, 2beta)`
`impliesr=alpha beta` and `alpha+beta=p` and `(alpha)/2+2beta=q`
`implies(beta=(2q-p)/3` and `alpha-(2(2p-q))/3`
`impliesalpha beta=r=2/9(2q-p)(2p-q)`