Question

The necessary and sufficient condition for the equation $(1-a^2)x^2+2ax-1=0$ to have roots lying in the interval $(0,1)$ is

• A. $a>0$
• B. $a<0$
• C. $a>2$
• D. none of these

Answer 1

The correct option is C $a>2$

$(1-a^2)x^2+2ax-1=0$

$x^2-(a^2{x}^2-2ax+1)=0$

$x^2-(ax-1{)}^2=0$

$(x+ax-1)(x-ax+1)=0$
$x=\frac{1}{1+a},\frac{1}{a-1}$ are the roots.
$0<\frac{1}{1+a}<1$
$\frac{1}{1+a}>0\Rightarrow 1+a>0\Rightarrow a>-1$............. (i)
$\frac{1}{1+a}<1\Rightarrow a>0$..................(ii)
And, $0<\frac{1}{a-1}<1$
$\frac{1}{a-1}>0\Rightarrow a-1>0\Rightarrow a>1$................(iii)
$\frac{1}{a-1}<1\Rightarrow a-1>1\Rightarrow a>2$............(iv)
On intersection of these four conditions, we get $a>2$.

This is the sufficient and necessary condition.
Hence, option 'C' is correct.