Let be roots of the equation, where. If assumes the minimum possible value, then is equal to _______ . [2024]

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Solution

Q.
Let $α, β \in N$ be roots of the equation $x^{2} - 70 x + λ = 0,$ where $\frac{λ}{2}, \frac{λ}{3} \notin N .$ If $λ$ assumes the minimum possible value, then $\frac{(\sqrt{α - 1} + \sqrt{β - 1}) (λ + 35)}{| α - β |}$ is equal to _______ . [2024]

Ans.
(60)

Given $α$ and $β$ are roots of $x^{2} - 70 x + λ = 0$

$\Rightarrow α + β = 70$ and $α β = λ$

We need to minimise $λ$

Let $f = α β = α (70 - α) .$

So, $f (α) = α (70 - α)$

0 and 70 are roots of $f .$

$f$ has a minimum value at $α = 0 or 70$

Now, $\frac{λ}{2}, \frac{λ}{3} \notin N$

It means $α$ and $β$ should not be multiples of 2 and 3.

$\Rightarrow For α = 1, β = 69$ which is a multiple of 3 similarly we can't take $α = 2, 3, 4 .$

So, for $α = 5, β = 65$

$\Rightarrow λ = α β = 325$ is the minimum value of $λ$ satisfying all the conditions.

Now, $\frac{(\sqrt{α - 1} + \sqrt{β - 1}) (λ + 35)}{| α - β |} = \frac{(\sqrt{4} + \sqrt{64}) (360)}{| - 60 |} = 60$

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