Let S = { log 2 ( 9 2 - 4 + 13 ) - log 2 ( 5 3 · 3 2 - 4 + 1 ) = 2 } . Then the maximum value of for which the equation x 2 - 2 ( S ) 2 x + S ( 1 ) 2 = 0 has real roots, is ________ . [2023]

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Solution

Q.
Let $S = {α : \log_{2} (9^{2 α - 4} + 13) - \log_{2} (\frac{5}{3} \cdot 3^{2 α - 4} + 1) = 2} .$ Then the maximum value of $β$ for which the equation

$x^{2} - 2 {(\sum_{α \in S} α)}^{2} x + \sum_{α \in S} {(α + 1)}^{2} β = 0$ has real roots, is ________ . [2023]

Ans.
(25)

$Given \log_{2} (9^{2 α - 4} + 13) - \log_{2} (\frac{5}{2} \cdot 3^{2 α - 4} + 1) = 2$

$\Rightarrow \log_{2} (\frac{9^{2 α - 4} + 13}{\frac{5}{2} \cdot 3^{2 α - 4} + 1}) = 2 \Rightarrow \frac{9^{2 α - 4} + 13}{\frac{5}{2} \cdot 3^{2 α - 4} + 1} = 4$

$\Rightarrow 9^{2 α - 4} + 13 = 10 \cdot 3^{2 α - 4} + 4 \Rightarrow 10 \cdot 3^{2 α - 4} - 9^{2 α - 4} = 9$

$Put α = 1, 10 \cdot 3^{- 2} - 9^{- 2} = \frac{10}{9} - \frac{1}{81} = \frac{89}{81} \neq 9 (not satisfy)$

$Put α = 2, 10 \cdot 3^{0} - 9^{0} = 10 - 1 = 9 (satisfy)$

$Put α = 3, 10 \cdot 3^{2} - 9^{2} = 90 - 81 = 9 (satisfy)$

$Hence, α = 2 or 3 .$

Now,

$\sum_{α \in S} α = 2 + 3 = 5$ and $\sum_{α \in S} {(α + 1)}^{2}$ $= \sum_{α \in S} α^{2} + 2 \sum_{α \in S} α + \sum_{α \in S} 1$

$= 4 + 9 + 2 (5) + 2 = 25$

So, $x^{2} - 2 {(5)}^{2} x + 25 β = 0$ $= x^{2} - 50 x + 25 β = 0$

For real roots, $D \geq 0$

$\Rightarrow 2500 - 4 (25 β) \geq 0 \Rightarrow 100 β \leq 2500 \Rightarrow β \leq 25$

$Hence, β_{\max} = 25$

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