The value of a for which the equation (1 − a^2 )x^2 + 2ax − 1 = 0 has roots belonging to (0, 1) is


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Question Number 20115 by Tinkutara last updated on 22/Aug/17

$The value of a for which the equation$ $(1 - a^{2}) x^{2} + 2 a x - 1 = 0 has roots$ $belonging to (0, 1) is$
	Answered by ajfour last updated on 22/Aug/17

	$r o o t s a r e r e a l a n d p r o d u c t o f r o o t s > 0$ $\Rightarrow a^{2} - 1 > 0$ $s u m o f r o o t s = - \frac{2 a}{(1 - a^{2})} > 0$ $k n o w i n g t h a t 1 - a^{2} < 0$ $w e c o n c l u d e a > 0$ $f (0) < 0, D ⩾ 0, f (1) < 0$ $\Rightarrow 4 a^{2} + 4 (1 - a^{2}) ⩾ 0; t r u e f o r a l l a$ $f (0) = - 1 < 0; a l w a y s t r u e$ $f (1) = 1 - a^{2} + 2 a - 1 = a (2 - a) < 0$ $\Rightarrow a < 0 o r a > 2$ $k n o w i n g a^{2} > 1 a n d a > 0$ $w e c a n s a y a \in (2, \infty) .$
	Commented by ajfour last updated on 22/Aug/17

	$y e s, t h a n k s .$
	Commented by Tinkutara last updated on 22/Aug/17

	$Thank you very much Sir!$
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