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Let $A$ be a square matrix of order 2 such that $|A|=2$ and the sum of its diagonal elements is -3 . If the points $(x, y)$ satisfying $\mathrm{A}^2+x \mathrm{~A}+y \mathrm{I}=\mathrm{O}$ lie on a hyperbola, whose $\text { length of semi major axis is } x \text { and semi minor axis is } y$, eccentricity is $\mathrm{e}$ and the length of the latus rectum is $l$, then $81\left(e^ 4+l^2\right)$ is equal to
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Verified Answer
The correct answer is:
233
Given $|A|=2$
trace $A=-3$
and $\mathrm{A}^2+\mathrm{xA}+\mathrm{yI}=0$
$\Rightarrow \mathrm{x}=3, \mathrm{y}=2$
$\begin{aligned}
& \text {so, } 81 e^{4}=169 \\
& L R=8 / 9 \\
& 81 e^{4}+81 L^{2}=169+64=233 \end{aligned}$
trace $A=-3$
and $\mathrm{A}^2+\mathrm{xA}+\mathrm{yI}=0$
$\Rightarrow \mathrm{x}=3, \mathrm{y}=2$
$\begin{aligned}
& \text {so, } 81 e^{4}=169 \\
& L R=8 / 9 \\
& 81 e^{4}+81 L^{2}=169+64=233 \end{aligned}$
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