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 _______. [2018]

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 _______. [2018]

Smart Achievers · Accepted Answer

(0.5)

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

$which has two roots α and β, such that α + β = \frac{π}{3}$

$∴$ $\sqrt{3} a \cos α + 2 b \sin α = c . . . (i)$

$and \sqrt{3} a \cos β + 2 b \sin β = c . . . (i i)$

$On subtracting equation (ii) from (i),$

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

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

$\Rightarrow - 2 \sqrt{3} a \sin \frac{π}{6} + 4 b \cos \frac{π}{6} = 0$ $(∵ \sin \frac{α - β}{2} \neq 0)$

$\Rightarrow - 2 \sqrt{3} a \times \frac{1}{2} + 4 b \frac{\sqrt{3}}{2} = 0$ $\Rightarrow \frac{b}{a} = \frac{1}{2} = 0.5$

Solution

Q.
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 _______. [2018]

Want Us to Email you About Special Offers & Updates?

Solution

Q. Let a,b,c be three non-zero real numbers such that the equation: 3acosx+2bsinx=c, x∈[-π2,π2], has two distinct real roots α and β with α+β=π3. Then, the value of ba is _______. [2018]

Want Us to Email you About Special Offers & Updates?

Q.
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 _______. [2018]