If alpha and beta are non-real, then con...

If `alpha and beta` are non-real, then condition for `x^(2) + alpha x + beta = 0` to have real roots, is

A

`(alpha-bar(alpha))(beta-bar(beta))=(alpha bar(beta)-bar(alpha)beta)^(2)`

B

`(bar(alpha)-alpha)(alphabar(beta)-bar(alpha)beta)=(beta-bar(beta))^(2)`

C

`(beta-bar(beta))(alpha bar(beta)-bar(alpha)beta)=(bar(alpha)-alpha)^(2)`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:

B

Let a be a real root of `x^(2) + alpha x + beta = 0`. Then,
`a^(2) + alpha a + beta = 0`
`a^(2) + bar(alpha) a + bar(beta) = 0`
`rArr" "(a^(2))/(alpha bar(beta)-bar(alpha)beta)=(a)/(beta-bar(beta))-(1)/(bar(alpha)-alpha)`
`rArr" "a=(alpha bar(beta)-bar(alpha)beta)/(beta-bar(beta))and a = (beta-bar(beta))/(bar(alpha)-alpha)`
`rArr" "(alpha bar(beta) - bar(alpha) beta)(bar(alpha)-alpha)=(beta-bar(beta))^(2)" "["On equating the values of a"]`

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