Let a,b,c be three non-zero real numbers such that the equation √3acosx+2bsinx=c, x∈ [- (π /2) , (π /2)] has two distinct real roots α β with α +β =(π /3) . Then the value of (4 b/a) is:

Question

Let a,b,c be three non-zero real numbers such that the equation √3acosx+2bsinx=c, x∈ [- (π /2) , (π /2)] has two distinct real roots α β with α +β =(π /3) . Then the value of (4 b/a) is: (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

√3cos x+(2 b/a)sin â¡ x=(c/a) Now, √3cos α +(2 b/a)sin â¡ α =(c/a) .....(i) √3cos β +(2 b/a)sin â¡ β =(c/a) .....(ii) √3[cos α - cos â¡ β ]+(2 b/a)(sin â¡ α - sin â¡ β )=0 √3[- 2 sin ((α + β /2)) sin â¡ ((α - β /2))]+(2 b/a)[2 cos â¡ ((α + β /2)) sin â¡ ((α - β /2))] =0 -√3+2√3.(b/a)=0 (4 b/a)=2

Q. Let a,b,c be three non-zero real numbers such that the equation 3​acosx+2bsinx=c,x∈[−2π​,2π​] has two distinct real roots α&β with α+β=3π​ . Then the value of a4b​ is:

3​cosx+a2b​sin⁡x=ac​ Now, 3​cosα+a2b​sin⁡α=ac​ .....(i) 3​cosβ+a2b​sin⁡β=ac​ .....(ii) 3​[cosα−cos⁡β]+a2b​(sin⁡α−sin⁡β)=0 3​[−2sin(2α+β​)sin⁡(2α−β​)]+a2b​[2cos⁡(2α+β​)sin⁡(2α−β​)] =0 −3​+23​.ab​=0 a4b​=2

Q. Let $a, b, c$ be three non-zero real numbers such that the equation $3 a cos x + 2 b s in x = c, x \in [- \frac{π}{2}, \frac{π}{2}]$ has two distinct real roots $α & β$ with $α + β = \frac{π}{3}$ . Then the value of $\frac{4 b}{a}$ is: