Let \( S \) be the set of all non-zero real numbers \( \alpha \) such that the quadratic equatio...

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Let \( S \) be the set of all non-zero real numbers \( \alpha \) such that the quadratic equation \( \alpha x^{2}-x+\alpha=0 \) has two distinct real roots \( x_{1} \) and \( x_{2} \) satisfying the inequality \( \left|x_{1}-x_{2}\right|1 \)
Which of the following intervals is (are) a subset(s) of \( S \) ?
[JEE Advanced 2015, 4M]
(a) \( \left(-\frac{1}{2},-\frac{1}{\sqrt{5}}\right) \)
(b) \( \left(-\frac{1}{\sqrt{5}}, 0\right) \)
(c) \( \left(0, \frac{1}{\sqrt{5}}\right) \)
(d) \( \left(\frac{1}{\sqrt{5}}, \frac{1}{2}\right) \)
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