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If alpha, beta are roots of the equation...

If `alpha, beta` are roots of the equation `2x^2 + 6x + b = 0 (b < 0),` then `alpha/beta+beta/alpha` is less than

A

2

B

-2

C

18

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
We have, `alpha+beta = -3 and alpha beta = (b)/(2)`
Since b `lt` 0, therefore discriminant `D = 36 - 4b gt 0`.
So, `alpha and beta` are real.
Now, `(alpha)/(beta)+(beta)/(alpha)=(alpha^(2)+beta^(2))/(alpha beta)=((alpha+beta)^(2)-2 alpha beta)/(alpha beta)=((alpha+beta)^(2))/(alpha beta)-2=(18)/(b)-2`
`rArr" "(alpha)/(beta) + (beta)/(alpha) lt -2" "[because b lt 0]`
ALITER We have, `alpha beta = (b)/(2) lt 0`
`therefore" "alpha and beta` are of opposite signs
`rArr" "(alpha)/(beta) lt 0`
`rArr" "(alpha)/(beta)+(beta)/(alpha) lt - 2" "[because x + (1)/(x) lt - 2 "for all x" lt 0, x ne -1]`
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