Let \(a, b, c\) be three non-zero real numbers such that the equation \(\sqrt{3} a \cos x+2 b \sin x=c\), \(x \in\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), has two distinct real roots \(\alpha\) and \(\beta\) with \(\alpha+\beta=\frac{\pi}{3}\). Then, the value of \(\frac{b}{a}\) is ____.

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Trigonometric Equations

Let

a, b, c

be three non-zero real numbers such that the equation

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 ____.

Answer: 0.5

The equation

3 a cos x + 2 b sin x = c

has roots

α

and

β = \frac{π}{3} - α

.

Substituting

β

3 a cos (\frac{π}{3} - α) + 2 b sin (\frac{π}{3} - α) = c

.

Expanding:

3 a (\frac{1}{2} cos α + \frac{3}{2} sin α) + 2 b (\frac{3}{2} cos α - \frac{1}{2} sin α) = c

.

Simplifying:

(\frac{3 a}{2} + b 3) cos α + (\frac{3 a}{2} - b) sin α = c

.

Comparing with the original equation

3 a cos α + 2 b sin α = c

\frac{3 a}{2} + b 3 = 3 a

and

\frac{3 a}{2} - b = 2 b

.

From the first:

b 3 = \frac{3 a}{2}

, so

b = \frac{a}{2}

.

Therefore

\frac{b}{a} = 0.5

JEE Advanced · 2018 · 20 May 2018 · Paper 1 · Numerical answer