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

Let `a ,\ b ,\ c` be three non-zero real numbers such that the equation `sqrt(3)\ acosx+2\ bsinx=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 `sqrt(3)a cos x + 2b sin x=c, x in [-pi/2, pi/2]`
`rArr sqrt(3)a ((1-t^(2))/(1+t^(2)))+2b ((2t)/(1+t^(2)))=c`, where `y=tan x/2`
`rArr sqrt(3) a (1-t^(2))+4bt=c(1+t^(2))`
`rArr t^(2) (c+sqrt(3) a)-4bt+c-sqrt(3)a=0`
Equation has roots `"tan" alpha/2, "tan" beta/2`
So, `"tan" alpha/2 +"tan" beta/2=(4b)/(c+sqrt(3) a)`
and `"tan" alpha/2 "tan" beta/2=(c-sqrt(3) a)/(c+sqrt(3) a)`
Now, `(alpha+beta)/2=pi/6`
`rArr tan ((alpha+beta)/2)=1/sqrt(3)`
`rArr ("tan"alpha/2+"tan" beta/2)/(1-"tan" alpha/2"tan" beta/2)=1/sqrt(3)`
`rArr (4b)/(c+sqrt(3)a-c+sqrt(3)a)=1/sqrt(3)`
`rArr b/a=1/2`
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