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  4. If the value of real number a>0 for which x2-5 a x+1-0 and x2-a x-5-0 have a common real root is (3/√2 β) then β is equal to

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

JEE MainJEE Main 2023Complex Numbers and Quadratic Equations

Solution:

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