Question

If the value of real number a> 0 for which \(x^2\)-5ax+1-0 and \(x^2-a x-5-0\) have a common real root is \(\frac{3}{\sqrt{2 \beta}}\) then \( \beta\) is equal to_______

Answer 1

Correct Answer: 13

Two equations have common root
$\therefore (4a)(26a)=(-6{)}^2=36$
$\Rightarrow a^2=\frac{9}{26}$
$\therefore a=\frac{3}{\sqrt{26}}$
$\Rightarrow \beta =13$
So, the correct answer is 13.

Answer 2

Let the two equations have a common root. \[ % Option (4a)(26a) = (-6)^2 = 36 \] \[ \Rightarrow a^2 = \frac{9}{26} \quad \Rightarrow a = \frac{3}{\sqrt{26}} \quad \Rightarrow \beta = 13 \]