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Q.

Let S be the set of all non-zero real numbers $α$ such that the quadratic equation $a x^{2} - x + α = 0$ has two distinct real roots $x_{1}$ and $x_{2}$ satisfying the inequality $| x_{1} - x_{2} | < 1.$ Which of the following intervals is(are) a subset(s) of S?

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a

$(- \frac{1}{2}, - \frac{1}{\sqrt{5}})$

b

$(0, \frac{1}{\sqrt{5}})$

c

$(- \frac{1}{\sqrt{5}}, \frac{1}{2})$

d

$(- \frac{1}{\sqrt{5}}, 0)$

answer is A.

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Detailed Solution

${(x_{1} - x_{2})}^{2} < 1 \Rightarrow \frac{1}{α^{2}} - 4 < 1$ and $1 - 4 α^{2} > 0$ .

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Let S be the set of all non-zero real numbers α such that the quadratic equation ax2−x+α=0has two distinct real roots x1 and x2 satisfying the inequality |x1−x2|<1. Which of the following intervals is(are) a subset(s) of S?