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Question #40833

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be the $3 \times 3$ symmetric matrix representing the rectangular hyperbola $H: x^2 - y^2 + 4xy + 6x - 8y + K = 0$ . A new matrix $N$ is obtained by applying the elementary row operation $R_3 \to R_3 + 4R_1 - 2R_2$ to $M$ . If the conic represented by the equation $\begin{pmatrix} x & y & 1 \end{pmatrix} N \begin{pmatrix} x \\ y \\ 1 \end{pmatrix} = 0$ degenerates into a pair of straight lines, then the value of $K$ is:

Options

A.
$-16$
B.
$-12$
C.
$4$
D.
$16$

Check Your Answer

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

Rectangular Hyperbola|Elementary Row Operations|Singular vs. Non-Singular Threshold

Why This Question Is Tricky

Common Mistakes

This question tests the structural understanding of conics via linear algebra.

The primary trap lies in the asymmetric nature of matrix $N$ after row operations.

Students often incorrectly assume $\det(N) = 0$ directly dictates degeneracy, forgetting that a quadratic form is governed strictly by the symmetric part of its matrix representation, $N_{sym} = \frac{N+N^T}{2}$ .

By forcing the expansion of $X^T N X = 0$ , the student must extract the new coefficients and evaluate the singular threshold (determinant of the new symmetric matrix) to find $K$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Rectangular Hyperbola
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free