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Question #74549

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be two skew lines in $\mathbb{R}^3$ with direction vectors $\vec{b}_1$ and $\vec{b}_2$ , passing through points with position vectors $\vec{a}_1$ and $\vec{a}_2$ respectively. Let $\vec{c} = \vec{a}_2 - \vec{a}_1$ . A $3 \times 3$ matrix $A$ is constructed such that its first, second, and third rows are the components of $\vec{c}$ , $\vec{b}_1$ , and $\vec{b}_2$ respectively. Let $C = \text{adj}(A A^T)$ be the adjugate matrix of $A A^T$ . If $c_{11}$ denotes the entry in the first row and first column of $C$ , which of the following expressions represents the shortest distance between the lines $L_1$ and $L_2$ ?

Options

A.
$\frac{(\det C)^{1/4}}{\sqrt{c_{11}}}$
B.
$\frac{\sqrt{\det C}}{c_{11}}$
C.
$\frac{(\det C)^{1/4}}{c_{11}}$
D.
$\frac{\sqrt{\det C}}{\sqrt{c_{11}}}$

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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

Shortest distance between skew lines|Scalar Triple Product Volume Logic|Determinant Product Rule

Why This Question Is Tricky

Examiner Levers

Variable Manipulation

Algebraic solutions only (No numbers); tests pure symbolic logic.

Difficulty Boost : +4

Common Mistakes

This question tests the synthesis of 3D geometry and linear algebra by obfuscating the standard shortest distance formula within matrix operations.

The primary trap lies in recognizing that the scalar triple product is the determinant of matrix $A$ , and forcing the use of the Determinant Product Rule to relate $\det(A)$ to the adjugate of the Gram matrix $A A^T$ .

Students commonly fail to identify that the cofactor $c_{11}$ of the Gram matrix represents Lagrange's identity for the squared magnitude of the cross product $|\vec{b}_1 \times \vec{b}_2|^2$ .

Furthermore, a frequent algebraic mistake is misapplying the adjugate determinant property, incorrectly assuming $\det(\text{adj}(M)) = \det(M)$ instead of $(\det M)^2$ for a $3 \times 3$ matrix, which leads to incorrect powers in the numerator.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Shortest distance between skew lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free