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Let a, b, c, be three non-zero real numb...

Let a, b, c, be three non-zero real numbers such that the equation `sqrt(3)a cos x+2b sin x=c, x in [-pi/2, pi/2]`, has two distinct real roots `alpha` and `beta` with `alpha+beta=pi/3`. Then, the value of `b/a` is ________.

Text Solution

Verified by Experts

The correct Answer is:
`(0.5)`
We have, `alpha, beta` are the roots of
`sqrt(3) a cos x + 2b sin x = c`
`therefore sqrt(3) a cos alpha + 2b sin alpha sin alpha = c" "…..(ii)`
and `sqrt(3) a cos beta + 2b sin beta= c " "……(i)`
On subtracting Eq. (ii) form Eq (i) we get
`sqrt(3)a (cos alpha -cos beta) + 2b (sin alpha - sinbeta) = 0`
`rArr sqrt(3)a(-2sin ((alpha + beta)/(2))) sin ((alpha +bet)/(2))+2b(2cos((alpha+beta)/(2)))sin ((alpha-beta)/(2))=0`
`rArr sqrt(3) a sin ((alpha+beta)/(2)) = 2b cos ((alpha+beta)/(2))`
`rArr tan ((alpha+ beta)/(2)) = (2b)/(sqrt(3a))`
`rArr tan((pi)/(6)) = (2b)/(sqrt(3a))[because alpha+ beta = (pi)/(3), given]`
`rArr (1)/(sqrt(3)) = (2b)/(sqrt(3a)) rArr (b)(a) = (1)/(2)`
`rArr (b)/(a) = 0.5`
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