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 \( A^2 + xA + yI = 0 \) 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 \( e^4 + \ell^4 \) is equal to _____

Answer 1

Correct Answer: 25

Given data: 
\[ |A| = 2, \quad \text{trace}(A) = -3 \] 
Matrix Equation: 
We are given: \[ A^2 + xA + yI = 0, \]
where \(I\) is the identity matrix. 

Interpreting the Condition: 
Since the given condition relates points \((x, y)\) that lie on a hyperbola whose transverse axis is parallel to the \(x\)-axis, we need to find the eccentricity \(e\) and the length of the latus rectum \(\ell\). 

Given Information: 
The problem states that \(|A| = 2\) and \(\text{trace}(A) = -3\). 

Using these conditions, we can establish that: \[ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix}, \]
where \(a + d = -3\) and \(ad - bc = 2\). 

Additional Conditions: 
Since the given problem does not provide sufficient information about the hyperbola’s parameters (such as the specific form of the matrix \(A\) or further constraints on \(x\) and \(y\)), determining the exact values of the eccentricity \(e\) and the latus rectum length \(\ell\) is not feasible. 

Conclusion: 
Based on the given conditions, the problem states an answer of \(e^4 + \ell^4 = 25\) as per the NTA’s answer key, but the derivation is incomplete due to insufficient data.