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Relationship between Zeroes and Coefficients of a Polynomial

If α, β are...

Question

If $α, β$ are real and distinct roots of $a x^{2} + b x - c = 0$ and $p, q$ are real and distinct roots of $a x^{2} + b x - | c | = 0$ , where $(a > 0)$ , then

A

α, β \in (p, q)

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B

α, β \in [p, q]

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C

p, q \in (α, β)

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D

None of these

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Solution

The correct option is A $α, β \in [p, q]$
$α, β$ are real and distinct roots of $a x^{2} + b x - c = 0$ .
Assume $α < β$
Therefore,
$α, β = \frac{- b \pm \sqrt{b^{2} + 4 a c}}{2 a}$
p, q be real and distinct roots of $a x^{2} + b x - | c | = 0$ .
Assume $p < q$
Therefore,
$p, q = \frac{- b \pm \sqrt{b^{2} + 4 a | c |}}{2 a}$
Now, $c \leq | c |$
$\Rightarrow 4 a c \leq 4 a | c |$
$\Rightarrow \sqrt{b^{2} + 4 a c} \leq \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow - b + \sqrt{b^{2} + 4 a c} \leq - b + \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow \frac{- b + \sqrt{b^{2} + 4 a c}}{2 a} \leq \frac{- b + \sqrt{b^{2} + 4 a | c |}}{2 a}$
$\Rightarrow β \leq q$
Also, $- \sqrt{b^{2} + 4 a c} \geq - \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow - b - \sqrt{b^{2} + 4 a c} \geq - b - \sqrt{b^{2} + 4 a | c |}$
$\Rightarrow \frac{- b - \sqrt{b^{2} + 4 a c}}{2 a} \geq \frac{- b - \sqrt{b^{2} + 4 a | c |}}{2 a}$
$\Rightarrow α \geq p$
Hence, $α, β \in [p, q]$ .

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