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Question #41749

Difficulty: 8/10 | 🟢 Free

The Question

Let $x \in \mathbb{R}$ such that the system of linear equations $\begin{pmatrix} x-1 & x-2 & x-3 \\ 2-x & x-1 & x-4 \\ 3-x & 4-x & x-1 \end{pmatrix} \begin{pmatrix} u \\ v \\ w \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \\ 0 \end{pmatrix}$ has a non-trivial solution. Let $L$ be a line with direction ratios $(u, v, w)$ . If the orthogonal projection of the line $L$ onto a plane $\Pi$ is a single point, and $\Pi$ passes through the point $A(2, 1, -1)$ , then the square of the perpendicular distance from the point $B(5, 2, 2)$ to the plane $\Pi$ is:

Options

A.
$\frac{25}{7}$
B.
$\frac{50}{7}$
C.
$\frac{100}{7}$
D.
$\frac{10}{7}$

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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

Orthogonal projection of a line|Collinearity vs Coplanarity|Skew-Symmetric Determinant|Factor Theorem in Determinants

Why This Question Is Tricky

Common Mistakes

This question tests the synthesis of algebraic determinant properties and geometric degeneracy.

Students often try to brute-force expand the $3 \times 3$ determinant, leading to a messy cubic equation prone to algebraic errors.

The trap relies on recognizing the matrix as a perturbed skew-symmetric matrix $(x-1)I + S$ .

By applying the Factor Theorem, students can immediately deduce that $(x-1)$ is a factor because setting $x=1$ leaves a purely skew-symmetric matrix of odd order, whose determinant is identically zero.

Furthermore, the geometric condition that the 'orthogonal projection of a line onto a plane is a point' is a conceptual trap; it simply means the line is perpendicular to the plane (degenerate projection), meaning the line's direction ratios become the plane's normal vector.

Failing to decode this geometric degeneracy prevents the formulation of the plane's equation.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Orthogonal projection of a line
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free