If 0 lt a lt b lt c and the roots alph...

If `0 lt a lt b lt c` and the roots `alpha,beta` of the equation `ax^2 + bx + c = 0` are non-real complex numbers, that

`|alpha|=|beta|`

`|alpha|gt1`

`|beta|lt1`

none of these

Text Solution

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The correct Answer is:

A, B

`0ltaltbltc, alpha+beta=(-b/a)` and `alpha beta=c/a`
For non real complex roots,
`b^(2)-4aclt0`
`implies(b^(2))/(a^(2))-(4c)/alt0`
`implies (alpha+beta)^(2)-4 alpha beta lt0`
`implies (alpha- beta)^(2)lt0`
`:' 0 lt a lt b lt c`
`:.` Roots are conjugate, then `|alpha|=|beta|`
But `alpha beta=c/a`
`|alpha beta|=|c/a|gt1[ :' a lt c , :. c/a gt1]`
`implies|alpha||beta|gt1`
`implies |alpha|^(2)gt1` or `|alpha|gt1`

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If 0 lt a lt b lt c and the roots alpha, beta of the equation ax ^(2)+bx+c=0 are imaginary, then

`|alpha |=|beta|`

`|alpha| gt 1`

`|beta| lt 1`

`|alpha |=1`

If 0 lt c lt b lt a and the roots alpha, beta the equation cx^(2)+bx+a=0 are imaginary, then

`(|alpha|+|beta|)/(2)=|alpha||beta|`

`(1)/(|alpha|)=(1)/(|beta|)`

`(1)/(|alpha|)+(1)/(|beta|)lt2`

`(1)/(|alpha|)+(1)/(|beta|)gt 2`

If c lt a lt b lt d , then roots of the equation bx^(2)+(1-b(c+d)x+bcd-a=0

are real and one lies between `c` and `a`

are real and distinct in which one lies between `a` and `b`

are real and distinct in which one lies between `c` and `d`

are not real