If every pair from among the equations `x^(2)+px+qr=0,andx^(2)+rx+pq=0` have a common root, then `(("sum of all distinct roots")/("Product of all distinct roots"))` is

Options

A. `2(p + q + r)`
B. `p + q + r`
C. `-(p + q + r)`
D. `pqr`

Answer

B

Explanation

The given equations are
`x^(2) + px + qr = 0" "...(i)`
`x^(2) + qx + rp = 0" "...(ii)`
and, `x^(2) + rx + pq = 0" "...(iii)`
Let `alpha, beta` be the roots of equation (i), `beta, gamma` be the roots of equation (ii) and `gamma, alpha` be the roots of equation (iii). Since `beta` is a common root of (i) and (ii).
`therefore" "beta^(2) p beta + qr = 0 and beta^(2) + q beta + rp = 0`
`rArr" "(p-q) beta + r(q-p) = 0 rArr beta = r`
Now, `alpha beta = qr rArr alpha r = qr rArr alpha = q`.
It is given that `beta and gamma` are roots of equation(ii).
`therefore" "beta gamma = rp rArr r gamma = rp rArr gamma = p`
Now, `alpha + beta + gamma - q + r + p = p + q + r` .
NOTE `alpha + beta + gamma` can also be equal to `-1//2(p+q+r) and 0`.