If `alpha, beta` are the roots of `x^(2) - px + q = 0` and `alpha', beta'` are the roots of `x^(2) - p' x + q' = 0`, then the value of `(alpha - alpha')^(2) + (beta + alpha')^(2) + (alpha - beta')^(2) + (beta - beta')^(2)` is

Options

A. `2{p^(2) - 2q + p'^(2) - 2q' - pp'}`
B. `2{p^(2) - 2q + p'^(2) - 2q' - qq'}`
C. `2{p^(2) - 2q - p'^(2) - 2q' + pp'}`
D. `2{p^(2) - 2q - p'^(2) - 2q' - qq'}`

Answer

1

Explanation

Since, `alpha, beta` are the roots of `x^(2) - px + q = 0`
and `alpha', beta'` are the roots of `x^(2) - p'x + q' = 0`, then
`alpha + beta = p, alphabeta = q, alpha' + beta' = p', alpha'beta' = q'`
Now, `(alpha - alpha')^(2) + (beta - alpha')^(2) + (alpha - beta')^(2) + (beta - beta')^(2)`
`= 2(alpha^(2) + beta^(2)) + 2(alpha'^(2) + beta'^(2) - 2 alpha' (alpha + beta) - 2beta' (alpha + beta)`
`= 2{(alpha + beta)^(2) - 2alphabeta + (alpha' + beta')^(2) - 2 alpha'beta' - (alpha + beta) (alpha' + beta')}`
`=2(p^(2) - 2q + p'^(2) - 2q' - pp')`