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

√(3)cos x+2b/asin x=c/a #160; ⇒√(3)cosα +2b/asinα =c/a #160; #160; ....... (1) ⇒√(3)cosβ +2b/asinβ =c/a #160; #160;........ ...

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Standard XII

Mathematics

Solving Simultaneous Trigonometric Equations

Let a, b, c b...

Question

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

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Solution

$\sqrt{3} cos x + \frac{2 b}{a} sin x = \frac{c}{a}$

$\Rightarrow \sqrt{3} cos α + \frac{2 b}{a} sin α = \frac{c}{a}$ ....... $(1)$

$\Rightarrow \sqrt{3} cos β + \frac{2 b}{a} sin β = \frac{c}{a}$ ........ $(2)$
Consider $(1) - (2)$
$\Rightarrow \sqrt{3} [cos α - cos β] + \frac{2 b}{a} (sin α - sin β) = 0$

$\Rightarrow \sqrt{3} [- 2 sin (\frac{α + β}{2}) sin (\frac{α - β}{2})] + \frac{2 b}{a} [2 cos (\frac{α + β}{2}) sin (\frac{α - β}{2})] = 0$

$\Rightarrow - \sqrt{3} + 2 \sqrt{3} \cdot \frac{b}{a} = 0$

$∴ \frac{b}{a} = \frac{1}{2} = 0.5$ .

$∴ \frac{2 b}{a} = 1$

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

√3cosx+2basinx=ca ⇒√3cosα+2basinα=ca .......(1)⇒√3cosβ+2basinβ=ca ........(2)Consider (1)−(2)⇒√3[cosα−cosβ]+2ba(sinα−sinβ)=0⇒√3[−2sin(α+β2)sin(α−β2)]+2ba[2cos(α+β2)sin(α−β2)]=0⇒−√3+2√3⋅ba=0∴ba=12=0.5.∴2ba=1

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