Let α, β are two real roots of equation x2+a x+b=0 (a, b ∈ R and b ≠ 0). If the quadratic equation g(x)=0 has two roots α+(1/α), β+(1/β) such that sum of roots is equal to product of roots, then the complete range of b is :

Question

Let α, β are two real roots of equation x2+a x+b=0 (a, b ∈ R and b ≠ 0). If the quadratic equation g(x)=0 has two roots α+(1/α), β+(1/β) such that sum of roots is equal to product of roots, then the complete range of b is : (A) [(1/3), 3] (B) ((1/3), 3] (C) [(1/3), 3) (D) (-∞, (1/3)) ∪(3, ∞)

Tardigrade Admin · Accepted Answer

α+β=-a ; α β=b

sum of roots = product of roots α+β+(1/α)+(1/β)=(α+(1/α))(β+(1/β)) ⇒ α+β+((α+β)/α β)=α β=(α/β)+(β/α)+(1/α β) Using equation (1) ⇒ -a-(a/b)=b+(a2-2 b/b)+(1/b) ⇒ a2+a(b+1)+(b2-2 b+1)=0 D ≥ 0 (Quadratic in a) (b+1)2-4(b2-2 b+1) ≥ 0 ⇒ (3 b-1)(b-3) ≤ 0 ⇒ b ∈[(1/3), 3]

Q. Let $α, β$ are two real roots of equation $x^{2} + a x + b = 0$ $(a, b \in R$ and $b \neq = 0)$ . If the quadratic equation $g (x) = 0$ has two roots $α + \frac{1}{α}, β + \frac{1}{β}$ such that sum of roots is equal to product of roots, then the complete range of b is :

$[\frac{1}{3}, 3]$

$(\frac{1}{3}, 3]$

$[\frac{1}{3}, 3)$

$(- \infty, \frac{1}{3}) \cup (3, \infty)$

Q. Let α,β are two real roots of equation x2+ax+b=0 (a,b∈R and b=0). If the quadratic equation g(x)=0 has two roots α+α1​,β+β1​ such that sum of roots is equal to product of roots, then the complete range of b is :

[31​,3]

(31​,3]

[31​,3)

(−∞,31​)∪(3,∞)

Q. Let $α, β$ are two real roots of equation $x^{2} + a x + b = 0$ $(a, b \in R$ and $b \neq = 0)$ . If the quadratic equation $g (x) = 0$ has two roots $α + \frac{1}{α}, β + \frac{1}{β}$ such that sum of roots is equal to product of roots, then the complete range of b is :

$[\frac{1}{3}, 3]$

$(\frac{1}{3}, 3]$

$[\frac{1}{3}, 3)$

$(- \infty, \frac{1}{3}) \cup (3, \infty)$