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Question #40817

Difficulty: 8/10 | 🟢 Free

The Question

Let $C_1, C_2,$ and $C_3$ be three conics given by the equations: $C_1: 2x^2 - 2xy + 2y^2 = 3$ $C_2: x^2 + y^2 = 3$ $C_3: ax^2 + 4xy + by^2 = 3$ For any variable line $L$ not passing through the origin, let its poles with respect to $C_1, C_2,$ and $C_3$ be $P_1, P_2,$ and $P_3$ respectively. If $P_1, P_2,$ and $P_3$ are collinear for all such lines $L$ , and $C_3$ is a non-degenerate conic, find the value of $a+b$ .

Options

A.
-2
B.
0
C.
2
D.
4

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

Polar and Pole Duality|Condition of Collinearity|Linear Independence vs Dependence

Why This Question Is Tricky

Common Mistakes

The problem tests the duality between poles and polars, specifically mapping the geometric condition of collinearity to the linear dependence of matrices.

The trap lies in attempting a brute-force coordinate geometry approach, which leads to an intractable $3 \times 3$ determinant of rational functions in $l, m, n$ .

The elegant 'Level 3' solution recognizes that the mapping from a line's normal vector to its pole is a linear transformation given by the inverse of the conic's coefficient matrix.

For the poles to be collinear for all lines, the inverse matrix of the third conic must be an affine combination (coefficients summing to 1) of the inverse matrices of the first two conics.

This bridges projective geometry with linear algebra, effectively separating top-tier candidates.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Conic Sections - Class 11
Concept	Polar and Pole Duality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free