Inequality

\hspace{-16}$If $\bf{xy=1}$ and $\bf{x,y\in\mathbb{R}}$ and satisfy the Relation $\bf{\left\{(x+y)^2+4\right\}.\left\{(x+y)^2-2\right\}\geq \mathbb{A}.(x-y)^2}$\\\\ Then $\bf{\mathbb{A}}$ is

3 Answers

21
Shubhodip ·

something like \frac{8 \sqrt{17} + 24}{ \sqrt{17} - 1} ?

36
rahul ·

let (x + y)2 = m => (x - y)2 = m - 4

given, (m + 4)(m - 2) > A (m - 4)

=> m2 + 2m - 8 - Am + 4A > 0

=> m2 + m (2 - A) - 8 + 4A > 0

clearly, a = coeff. of m > 1

=> D = 0

=> (2 - A)2 - 4 (4A - 8) = 0

=> 4 + A2 - 4A - 16A + 32 = 0

=> A2 - 20A + 36 = 0

=> A = 18 or 2 ???????

1708
man111 singh ·

Thanks Rahul

Answer Given is = 18

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