Let a , b , c be three non-zero real numbers such that the equation 3 a cos ⁡ x + 2 b sin ⁡ x = c , x ∈ - π 2 , π 2 has two distinct real roots α and β with α + β = π 3 . Then, the value of b a is

Step-by-step solution

3 cos ⁡ x + 2 b a sin ⁡ x = c a Now, 3 cos ⁡ α + 2 b a sin ⁡ α = c a   … i 3 cos ⁡ β + 2 b a sin ⁡ β = c a   … ii From equations i and ii ⇒ 3 cos ⁡ α - cos ⁡ β + 2 b a sin ⁡ α - sin ⁡ β = 0 ⇒ 3 - 2 sin ⁡ α + β 2 sin ⁡ α - β 2 + 2 b a 2 cos ⁡ α + β 2 sin ⁡ α - β 2 = 0 ⇒ - 3 + 2 3 · b a = 0 ⇒ b a = 1 2 = 0