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Question #40733

Difficulty: 8/10 | 🟢 Free

The Question

Consider the curve $C: |y| = x^2 - x$ . Let $\Gamma_1$ be the parabola $y = x^2 - x$ and $\Gamma_2$ be the parabola $y = x - x^2$ . For a variable line $L$ , let $P_1$ and $P_2$ be the poles of $L$ with respect to $\Gamma_1$ and $\Gamma_2$ respectively. Let $S$ be the set of all lines $L$ such that $P_1$ lies on $\Gamma_2$ and $P_2$ lies on $\Gamma_1$ . If the lines in $S$ enclose a bounded region with the $y$ -axis, find the area of this region.

Options

A.
$\frac{\sqrt{3}}{4}$
B.
$\frac{\sqrt{3}}{2}$
C.
$\frac{3\sqrt{3}}{4}$
D.
$\sqrt{3}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Monster

Ultra-high difficulty questions designed to test your limit, stamina, and deep conceptual understanding.

Concepts Behind This Question

Slope Sensitivity|Polar and Pole Duality|Modulus Function Symmetry

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Function Symmetry' trap combined with 'Slope Sensitivity' in pole-polar duality.

Students often incorrectly attempt to find the pole with respect to the modulus equation directly, which is algebraically invalid.

The modulus forces the definition of two distinct parabolas.

When finding the polar relative to these branches, the shift in the $y$ -coefficient's sign across regions creates a slope sensitivity where the $x$ -coordinate of the pole shifts from $(m+1)/2$ to $(1-m)/2$ .

The cross-branch locus condition ( $P_1 \in \Gamma_2$ and $P_2 \in \Gamma_1$ ) forces students to maintain strict sign conventions.

Equating the $y$ -intercepts locks in the specific slopes, rewarding those who correctly navigated the piecewise algebraic geometry.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Slope Sensitivity
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free