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Question #40790

Difficulty: 8/10 | 🟢 Free

The Question

Let $M$ be the $3 \times 3$ symmetric matrix associated with the conic $S \equiv 2xy + 4x + 6y = 0$ , where the conic is represented as $\mathbf{x}^T M \mathbf{x} = 0$ with $\mathbf{x} = \begin{pmatrix} x \\ y \\ 1 \end{pmatrix}$ . Let $C$ be the cofactor matrix of $M$ . A line $L$ is defined by the equation $\begin{pmatrix} x & y & 1 \end{pmatrix} C \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} = 0$ . If $(x_0, y_0)$ is the pole of the line $L$ with respect to the conic $S=0$ , then find the coordinates $(x_0, y_0)$ .

Options

A.
$(-9, 7)$
B.
$(-1, 1)$
C.
$(-9, -7)$
D.
$(3, -2)$

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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

General Conic Definition|Polar and Pole Duality|Cramer's Rule Constraints|Row/Column Exchange Parity

Why This Question Is Tricky

Common Mistakes

This question tests the duality between poles and polars using the matrix representation of a general conic, specifically targeting the row/column exchange parity trap in determinant and cofactor calculations.

Students must first extract the symmetric matrix $M$ and compute its cofactor matrix $C$ to identify the line $L$ .

The primary trap lies in the calculation of the off-diagonal cofactors (e.g., $C_{12}$ and $C_{32}$ ); students frequently neglect the $(-1)^{i+j}$ parity factor, leading to an incorrect line equation (like $3x + 2y + 1 = 0$ ).

Finding the pole of this false line yields the distractor $(-1, 1)$ .

Furthermore, applying the false cofactor matrix again to find the pole leads to another distractor $(-9, -7)$ .

Mastery requires strict adherence to Cramer's rule constraints and cofactor signs.

Quick Facts About This Question

Question Information

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