JEE MAIN - Mathematics (2024 - 4th April Morning Shift - No. 23)
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 transverse axis is parallel to the $$x$$-axis, eccentricity is $$\mathrm{e}$$ and the length of the latus rectum is $$l$$, then $$\mathrm{e}^4+l^4$$ is equal to ________.
Answer
25
Explanation
$$|A|=2 \sum \mathrm{dia}=-3$$
$$\therefore \quad$$ character equation : $$A^2+3 A+2 I=0$$
$$\Rightarrow x=3 \quad y=2$$
$$\because$$ We are getting only one point $$(3,2)$$ but its given many points satisfy this equation.
Moreover hyperbola whose transverse axis is $$x$$ axis and passing through $$(3,2)$$ is not unique.
$$\therefore$$ multiple value of '$$e$$' and $$L(L R)$$ is possible.
We'll not get a unique result.
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