If - - are the roots of 2x2-6x-b-0 and -2 then b lies in the interval

Solution : α, β are roots of 2x^2+6x+b=0 α+β = 3 αβ =b/2 α/β+β/α 2 α^2+β^2/αβ 2 (α+β)^2 2αβ/αβ 2 (α+β)^2/αβ 4 9/(b/2) 4 1/b2/9 b ∈ ( ∞, 0) ∪(9/2,∞) ...

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Standard XII

Mathematics

Relation between Roots and Coefficients for Quadratic

If α, β are t...

Question

If $α, β$ are the roots of $2 x^{2} + 6 x + b = 0$ and $\frac{α}{β} + \frac{β}{α} < 2$ then b lies in the interval

(0, \infty)

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(- 1, \infty)

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(- \infty, 0)

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(- \infty, 0]

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Solution

The correct option is C $(- \infty, 0)$
Solution : $α, β$ are roots of $2 x^{2} + 6 x + b = 0$
$α + β = - 3$
$α β = \frac{b}{2}$
$\frac{α}{β} + \frac{β}{α} < 2$
$\frac{α^{2} + β^{2}}{α β} < 2$
$\frac{(α + β)^{2} - 2 α β}{α β} < 2$
$\frac{(α + β)^{2}}{α β} < 4$
$\frac{9}{(\frac{b}{2})} < 4$
$\frac{1}{b} < \frac{2}{9}$
$b \in (- \infty, 0) \cup (\frac{9}{2}, \infty)$

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If α, β are the roots of 2x2+6x+b=0 and αβ+βα<2 then b lies in the interval

The correct option is C (−∞,0)Solution : α, β are roots of 2x2+6x+b=0 α+β=−3 αβ=b2 αβ+βα<2 α2+β2αβ<2 (α+β)2−2αβαβ<2 (α+β)2αβ<4 9(b2)<4 1b<29 b∈(−∞, 0)∪(92,∞)

If $α, β$ are the roots of $2 x^{2} + 6 x + b = 0$ and $\frac{α}{β} + \frac{β}{α} < 2$ then b lies in the interval