If `alpha` and `beta(alpha lt beta)` are the roots of the equation `x^(2)+bx+c=0` where `c lt 0ltb`, then

Options

A. `0 lt alpha lt beta`
B. `alpha lt 0 beta lt |alpha|`
C. `alpha lt beta lt 0`
D. `alpha lt 0 lt |alpha| lt beta`

Answer

B

Explanation

Since `alpha, beta` are the roots of the equation `x^(2) + bx + c = 0`.
`therefore" "alpha+beta = -b and alpha beta = c`
Now, `c lt 0 lt b`
`rArr" "alpha beta lt 0 and alpha+beta lt 0`
`rArr" "`One of the roots is positive and other is negative.
It is given that `alpha lt beta`. Therefore, `alpha lt 0 and beta gt 0`
Now, `alpha+beta = -b`
`rArr" "beta = -b - alpha`
`rArr" "beta lt - alpha" "[because b gt 0]`
`rArr" "beta lt |alpha|" "[because alpha lt 0, therefore |alpha| = - alpha]`
Thus, `alpha lt 0 lt beta lt |alpha|`.