Mathematics - Quadratic Equation Question with Solution | TestHub

$\left|x_1+x_2\right|\left|x_1-x_2\right|>\frac{1}{a}\Rightarrow \left|-\frac{1}{a}\right|\left|\sqrt{\frac{1}{a^2}-4}\right|>\frac{1}{a}\cdot \left({}^{\ast }\right)$

The inequation (\textit{) has meaning if $\frac{1}{a^2}-4>0$

If $a\in \left(-\frac{1}{2},0\right)$ then (}) is automatically satisfied

If $a\in \left(0,\frac{1}{2}\right)$ then $\left({}^{\ast }\right)$ becomes equivalent to $\sqrt{\frac{1}{a^2}-4}>1$ (on canceling $\frac{1}{a}>0$ ) $\Rightarrow -\frac{1}{\sqrt{5}}<a<\frac{1}{\sqrt{5}}$

but $a\in \left(0,\frac{1}{2}\right)$ was assumed $\Rightarrow a\in \left(0,\frac{1}{\sqrt{5}}\right)$

Thus all the values of $a$ lie in the interval $\left(-\frac{1}{2},0\right)\cup \left(0,\frac{1}{\sqrt{5}}\right)\Rightarrow k=5$