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Question

# The necessary and sufficient condition for the equation (1âˆ’a2)x2+2axâˆ’1=0 to have roots lying in the interval (0,1) is

A
a>0
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B
a<0
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C
a>2
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D
none of these
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Solution

## The correct option is C a>2(1−a2)x2+2ax−1=0x2−(a2x2−2ax+1)=0x2−(ax−1)2=0(x+ax−1)(x−ax+1)=0x=11+a,1a−1 are the roots.0<11+a<1 11+a>0⇒1+a>0⇒a>−1............. (i)11+a<1⇒a>0..................(ii)And, 0<1a−1<11a−1>0⇒a−1>0⇒a>1................(iii)1a−1<1⇒a−1>1⇒a>2............(iv)On intersection of these four conditions, we get a>2.This is the sufficient and necessary condition.Hence, option 'C' is correct.

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