The-value-of-a-for-which-the-equation-1-a-2-x-2-2ax-1-0-has-roots-belonging-to-0-1-is-

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!