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Question #40834

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = 3\hat{i} + \hat{j}$ , $\vec{b} = \hat{i} + 5\hat{j}$ , and $\vec{c} = 14\hat{i} + 18\hat{j}$ . A conic section $H$ in the $xy$ -plane is defined as the locus of a point $P$ with position vector $\vec{r} = x\hat{i} + y\hat{j}$ satisfying the vector equation: $(\vec{r} \times \vec{a}) \cdot (\vec{b} \times \vec{r}) + 4|\vec{r}|^2 = \vec{r} \cdot \vec{c}$ If $\vec{r}_c$ is the position vector of the center of $H$ , find the value of $\vec{r}_c \cdot (2\hat{i} - \hat{j})$ .

Options

A.
1
B.
-105
C.
25
D.
2

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Rectangular Hyperbola|Vector Product Anti-Commutativity|Position Vector Origin Dependence

Why This Question Is Tricky

Common Mistakes

This question tests the student's rigorous handling of vector operations within coordinate geometry, specifically targeting the anti-commutativity of the cross product ( $A \times B = -B \times A$ ).

A common trap is to carelessly rewrite $(\vec{b} \times \vec{r})$ as $(\vec{r} \times \vec{b})$ without flipping the sign, which drastically alters the quadratic terms of the resulting conic section (leading to the false center $(-121, -137)$ and option -105).

Additionally, it punishes students who mistakenly use dot products instead of cross products (leading to option 25), ensuring only geometrically and algebraically precise execution yields the correct center.

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