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 b-a is -

√(3) a cos x + 2 b sin x = c and α + β = π/3 Substituting x = α and x = β, √(3) a cosα + 2 b sinα = c..................(i) √(3) a cosβ + 2 b sinβ = c........... ...

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Standard XII

Mathematics

Solving Simultaneous Trigonometric Equations

Let a , b , c...

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

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

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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 .