Let α, β are real roots of equation x2 + ax + b = 0 ; where a R and b 0. If another equation has two roots and such that sum of its roots is equal to product of roots, then range of b is [m, M]. Find the value of mM.

Engineering

Mathematics

maximum and minimum of quadratic and rational expression

Question

Let α, β are real roots of equation x² + ax + b = 0 ; where a $\in$ R and b $\neq$ 0. If another equation has two roots $(α + \frac{1}{α})$ and $(β + \frac{1}{β})$ such that sum of its roots is equal to product of roots, then range of b is [m, M]. Find the value of mM.

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$α + \frac{1}{α} + β + \frac{1}{β} = (α + \frac{1}{α}) (β + \frac{1}{β})$

⇒ $- a - \frac{a}{b} = b + \frac{1}{b} + \frac{a^{2} - 2 b}{b}$

⇒   b² + 1 + a² – 2b + ab + a = 0
⇒   a² + (b + 1)a + (b – 1)² = 0
$∴$ D $\geq$ 0
⇒   (b + 1)² – 4(b – 1)² $\geq$ 0
⇒   b² + 2b + 1 – 4b² + 8b – 4 $\geq$ 0
⇒   3b² – 10b + 3 $\leq$ 0   ⇒ $b \in [\frac{1}{3}, 3]$ ⇒ mM = 1