Let sin x and sin y be roots of the quadratic equation sin 2 θ+b sin θ+c=0(a, b, c ∈ R and a ≠ 0) such that sin x+2 sin y=1, then the value of (a2+2 b2+3 a b+a c) equals

Question

Let sin x and sin y be roots of the quadratic equation sin 2 θ+b sin θ+c=0(a, b, c ∈ R and a ≠ 0) such that sin x+2 sin y=1, then the value of (a2+2 b2+3 a b+a c) equals (A) 0 (B) 1 (C) 2 (D) 4

Tardigrade Admin · Accepted Answer

Given ( sin x+ sin y)=(-b/a) ....(1) and sin x ⋅ sin y=(c/a) ....(2) also sin x+2 sin y=1 ....(3) From (1) and (3) ⇒ sin y=1+(b/a)=(a+b/a) ⇒ sin x=(-b/a)-((a+b)/a)=(-a-2 b/a) So, put in (2), we get (-( a +2 b )/ a ) ⋅ (( a + b )/ a )=( c / a ) Hence, a c+(a+2 b)(a+b)=0 ⇒ a2+2 b2+3 a b+a c=0

Q. Let sinx and siny be roots of the quadratic equation sin2θ+bsinθ+c=0(a,b,c∈R and a=0) such that sinx+2siny=1, then the value of (a2+2b2+3ab+ac) equals

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Given (sinx+siny)=a−b​ ....(1) and sinx⋅siny=ac​ ....(2) also sinx+2siny=1 ....(3) From (1) and (3) ⇒siny=1+ab​=aa+b​ ⇒sinx=a−b​−a(a+b)​=a−a−2b​ So, put in (2), we get a−(a+2b)​⋅a(a+b)​=ac​ Hence, ac+(a+2b)(a+b)=0⇒a2+2b2+3ab+ac=0

Q. Let $sin x$ and $sin y$ be roots of the quadratic equation $sin^{2} θ + b sin θ + c = 0 (a, b, c \in R$ and $a \neq = 0)$ such that $sin x + 2 sin y = 1$ , then the value of $(a^{2} + 2 b^{2} + 3 ab + a c)$ equals