If `alpha,beta` are the roots of `x^2+p x+q=0a n dgamma,delta` are the roots of `x^2+p x+r=0,` then `((alpha-gamma)(alpha-delta))/((beta-gamma)(beta-delta))=` `1` b. `q` c. `r` d. `q+r`

Options

A. 1
B. q
C. r
D. q + r

Answer

1

Explanation

`alpha, beta` be the roots of `x^(2) + px + q = 0`
`gamma, delta` be the roots of `x^(2) + px + r = 0`
`alpha + beta = -p`
`alphabeta = q`
`gamma + delta = -p`
`gamma delta = r`
Now, `(alpha - gamma)(alpha - delta) = alpha^(2) - alpha(gamma + delta) + gammadelta`
`= alpha^(2) - alpha(alpha + beta) = r`
`= - alphabeta + r = -q + r`
By symmetry `(beta - gamma) (beta - delta) = -q + r`
Hence, `((alpha - gamma)(alpha - delta))/((beta - gamma)(beta - delta)) = 1`