Let a,b,c be the sides of a triangle, a≠ b≠ c, λ∈ R. If the roots of x²+2(a+b+c)x+3λ(ab+bc+ca)=0 are real, then (a) λ 53 (c) λ∈ ( 13, 53 ) (d) λ∈ ( 43, 53 )

Correct answer: (a) λ< 43 Real roots need (a+b+c)² 3λ(ab+bc+ca), i.e. λ≤((a+b+c)²)/(3(ab+bc+ca)); for triangle sides this ratio is < 43, so λ< 43. JEE Advanced 2006 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.

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[JEE Advanced 2006] let a b c be the sides of a triangle a b c lambda

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Let $a,b,c$ be the sides of a triangle, $a\ne b\ne c$, $\lambda\in\mathbb R$. If the roots of $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real, then

(a) $\lambda<\frac43$
(b) $\lambda>\frac53$
(c) $\lambda\in\left(\frac13,\frac53\right)$
(d) $\lambda\in\left(\frac43,\frac53\right)$

1 Answer

VAVidaara Admin ✓ Vidaara Team ✓ Accepted · 2d ago ▲ 0

Correct answer: (a) $\lambda<\frac43$

Real roots need $(a+b+c)^2\ge3\lambda(ab+bc+ca)$, i.e. $\lambda\le\frac{(a+b+c)^2}{3(ab+bc+ca)}$; for triangle sides this ratio is $<\frac43$, so $\lambda<\frac43$.

JEE Advanced 2006 · Quadratic Equations and Inequations — verified solution by the Vidaara Team.

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