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Question #40776

Difficulty: 8/10 | 🟢 Free

The Question

Let $x, y, z$ be distinct positive real numbers, and let $A = \begin{pmatrix} \ln x & \ln y & \ln(x/y) \\ \ln y & \ln z & \ln(y/z) \\ \ln z & \ln x & \ln(z/x) \end{pmatrix}$ Suppose the characteristic equation of $A$ is given by $\lambda^3 - c_1\lambda^2 + c_2\lambda - c_3 = 0$ , and let its roots $\lambda_1, \lambda_2, \lambda_3$ be distinct. Consider the three lines $L_1, L_2, L_3$ in the $xy$ -plane defined by: $L_i : x + \lambda_i y + c_3 \cos(\lambda_i) = 0 \quad \text{for } i = 1, 2, 3$ The area of the triangle formed by these three lines is:

Options

A.
0
B.
$\frac{1}{2} |c_1 c_2|$
C.
$\frac{1}{2} |c_1^2 - 2c_2|$
D.
1

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Bridge

Practice problems that bridge the gap between foundational theory and advanced practical application.

Concepts Behind This Question

Area of Triangle via Determinants|Concurrent Lines Condition|Characteristic Equation

Why This Question Is Tricky

Common Mistakes

This question tests the integration of linear algebra and coordinate geometry, specifically targeting the conceptual link between a matrix's determinant, its characteristic equation, and the geometric condition for concurrent lines.

The trap lies in the constant term of the line equations, $c_3 \cos(\lambda_i)$ .

Students who blindly attempt to compute the intersection points and apply the determinant formula for the area of a triangle will be bogged down in an intractable calculation.

The conceptual filter requires recognizing that the third column of the matrix is the difference of the first two, making the matrix singular ( $\det A = 0$ ).

Consequently, the constant term $c_3$ in the characteristic equation is zero, reducing the lines to $x + \lambda_i y = 0$ .

These lines are trivially concurrent at the origin, forcing the triangle's area to zero.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Area of Triangle via Determinants
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free