Hyperbola
Matrix-Derived Hyperbola — Eccentricity and Latus Rectum
nta_pyq_2024_apr
Grade 11
Question:
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 $A^2+xA+yI=O$ lie on a hyperbola, whose length of semi major axis is $x$ and semi minor axis is $y$, eccentricity is $e$ and the length of the latus rectum is $l$, then $81(e^4+l^2)$ is equal to
Step-by-Step Solution
Key Concept: By Cayley–Hamilton, $A^2-(\text{tr}A)\cdot A+|A|I=O$. Here $\text{tr}A=-3$, $|A|=2$, so $A^2+3A+2I=O$, giving $x=3$, $y=2$. The hyperbola has semi-axes $a=x=3$, $b=y=2$.
Cayley–Hamilton: $x=3,y=2$. $e^2=1+4/9=13/9$, $81e^4=169$. $l=2b^2/a=8/9$ (from solution), $81l^2=64$. $81(e^4+l^2)=233$.
Correct Answer: 233