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 3 2 then is equal to _______ . [2023]

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Solution

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 $\frac{3}{\sqrt{2 β}}$ then $β$ is equal to _______ . [2023]

Ans.
(13)

$x^{2} - 5 a x + 1 = 0$ ...(i)

$Here, a_{1} = 1, b_{1} = - 5 a, c_{1} = 1$

   $x^{2} - a x - 5 = 0$ ...(ii)

$a_{2} = 1, b_{2} = - a, c_{2} = - 5$

$∵ Equation (i) and (ii) have one common real root, so$

   ${[1 \times 1 - (- 5) (1)]}^{2} = [(- 5 a) (- 5) - (- a) (1)] (- a + 5 a)$

$\Rightarrow 36 = (25 a + a) (4 a) \Rightarrow 36 = 26 a \times 4 a$

$∴$    $a^{2} = \frac{36}{26 \times 4} \Rightarrow a = \pm \frac{3}{\sqrt{26}}$

Since, the value of the root is given as $\frac{3}{\sqrt{2 β}}$ .

So,   $β = 13$

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