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Question #40816

Difficulty: 8/10 | 🟢 Free

The Question

Let $\Gamma$ be a family of curves in the first quadrant of the $xy$ -plane. For any point $P(x,y)$ on a curve in $\Gamma$ , consider the conic $C_P$ given by the equation $X^2 + \frac{y}{x} X Y + Y^2 = 1$ in the $XY$ -plane. The polar of $P(x,y)$ with respect to $C_P$ is identical to the tangent line to the orthogonal trajectory of $\Gamma$ passing through $P$ . If the curve in $\Gamma$ passes through the point $(\sqrt{2}, 1)$ , then the value of $y^2$ when $x = 2\sqrt{3}$ is:

Options

A.
4
B.
6
C.
8
D.
10

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|Orthogonal Trajectories|Tangent/Normal Slope Interplay

Why This Question Is Tricky

Common Mistakes

This question tests the interplay between pole-polar duality and orthogonal trajectories, specifically targeting slope relationships.

The primary trap lies in the interpretation of 'identical to the tangent line to the orthogonal trajectory'.

Students often mistakenly equate the slope of the polar directly to the slope of the curve $\Gamma$ ( $dy/dx = m_{polar}$ ), or apply the negative reciprocal incorrectly, forgetting that the orthogonal trajectory's tangent already incorporates the $-1/m$ condition.

This leads to an incorrect homogeneous differential equation.

Furthermore, the problem requires setting up a parameter-dependent conic $C_P$ and correctly applying the polar equation $T=0$ with current coordinates, filtering out candidates who lack rigorous algebraic control.

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