Let a- b- c be three non zero real numbers such that the equation-3 a cos x - 2 b sin x - c- x -2- -2-has two distinct real roots - and - with - - - - -3-Then- the value of ba is -

Given:√(3) a cos x + 2 b sin x = c and α + β = π3 Substituting x = α and x = β, √(3) a cosα + 2 b sinα = c⋯⋯(1) √(3) a cosβ + 2 b sinβ = c ⋯⋯(2) Subtracting eq ...

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Let a, b, c b...

Question

$Let a, b, c be three non zero real numbers such that the equation$
$\sqrt{3} a cos x + 2 b sin x = c, x \in [- \frac{π}{2}, \frac{π}{2}]$
$has two distinct real roots α and β with α + β = \frac{π}{3} .$
$Then, the value of \frac{b}{a} is$
.

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Solution

$Given : \sqrt{3} a cos x + 2 b sin x = c$ and
$α + β = \frac{π}{3}$
$Substituting x = α and x = β$ ,
$\sqrt{3} a cos α + 2 b sin α = c \dots \dots (1)$
$\sqrt{3} a cos β + 2 b sin β = c \dots \dots (2)$
Subtracting equation (2) from (1),
$\sqrt{3} a (cos α - cos β) + 2 b (sin α - sin β) = 0 \Rightarrow \sqrt{3} a [- 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})] + 2 b [2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2})] = 0 using α + β = \frac{π}{3},$
$\Rightarrow \sqrt{3} a [- sin (\frac{α - β}{2})] + 2 b [\sqrt{3} sin (\frac{α - β}{2})] = 0$
$∴ - \sqrt{3} a + 2 \sqrt{3} b = 0 \Rightarrow \frac{b}{a} = \frac{\sqrt{3}}{2 \sqrt{3}} \Rightarrow \frac{b}{a} = 0.5$

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Let a,b,c be three non zero real numbers such that the equation √3 acosx+2 bsinx=c, x∈[−π2,π2] has two distinct real roots α and β with α+β=π3. Then, the value of ba is .

$Let a, b, c be three non zero real numbers such that the equation$
$\sqrt{3} a cos x + 2 b sin x = c, x \in [- \frac{π}{2}, \frac{π}{2}]$
$has two distinct real roots α and β with α + β = \frac{π}{3} .$
$Then, the value of \frac{b}{a} is$
.