The values of a for which( $a^{2} - 1$ ) $x^{2}$ + 2(a-1)x + 2 is positive for any x are

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Solution

The correct option is D

We know that the expression a $x^{2}$ + b $x$ + c > 0 for all $x$ , if a > 0 and $b^{2}$ < 4ac

∴ $(a^{2} - 1)$ $x^{2}$ + 2(a-1) $x$ + 2 is positive for all $x$ , if $a^{2} - 1$ > 0 and 4 $(a - 1)^{2} - 8 (a^{2} - 1)$ < 0

⇒ $a^{2} - 1$ > 0 and -4(a - 1)(a + 3) < 0

⇒ $a^{2} - 1$ > 0 and (a - 1)(a + 3) > 0

⇒ $a^{2}$ > 1 and a < -3 or a > 1

⇒ a < -3 or a > 1

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