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Question #40770

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_H$ be the line $x = y = z$ and $F$ be the family of planes containing $L_H$ . Let $L_0$ be a line given by $\frac{x-1}{a} = \frac{y-2}{b} = \frac{z-3}{c}$ . For each plane $\Pi \in F$ , let $L_\Pi$ be the orthogonal projection of $L_0$ onto $\Pi$ . It is given that the lines $L_\Pi$ are concurrent at a single point, but $L_0$ and $L_H$ are skew lines. Which of the following is a valid direction vector $\langle a, b, c \rangle$ for $L_0$ ?

Options

A.
$\langle 2, -3, 1 \rangle$
B.
$\langle 1, 2, 3 \rangle$
C.
$\langle 3, 1, -1 \rangle$
D.
$\langle 1, -1, 1 \rangle$

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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

Family of Lines|Equation of plane through intersection|Orthogonal projection of a line

Why This Question Is Tricky

Common Mistakes

This question tests the spatial visualization and algebraic conditions for the orthogonal projection of a line onto a pencil of planes.

The core trap lies in the assumption that for the projections to be concurrent, the original line must intersect the hinge line of the family of planes.

Students often erroneously set up the intersection condition ( $a-2b+c=0$ ) and select a trap option.

However, the problem explicitly states the lines are skew.

The mathematical collapse happens when students fail to realize that orthogonal projection preserves concurrency if the projection vector (normal to the plane) maintains a specific relationship with the hinge line, which occurs when the original line is strictly orthogonal to the hinge line ( $a+b+c=0$ ).

This requires mastery of 3D coordinate geometry and vector triple products.

Quick Facts About This Question

Question Information

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