Let A be a square matrix of order 2 such that |A|=2 and sum of its diagonal elements is -3 . If poin

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3 and 4 .Determinants and Matrices

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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 $A^2+x A+y I=0$ lie on a hyperbola, whose transverse axis is parallel to the x-axis, eccentricity is e and the length of the latus rectum is $\ell$, then $\mathrm{e}^4+\ell^4$ is equal to...........................

A

$25$

B

$78$

C

$28$

D

$46$

(JEE MAIN-2024)

Solution

Given $|A|=2$

trace $\mathrm{A}=-3$

and $\mathrm{A}^2+\mathrm{xA}+\mathrm{yI}=0$

$\Rightarrow \mathrm{x}=3, \mathrm{y}=2$

so, information is incomplete to determine eccentricity of hyperbola ($e$) and length of latus rectum of hyperbola $(\ell)$

Standard 12

Mathematics

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(JEE MAIN-2015)