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Question #73278

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ and $L_2$ be lines in $\mathbb{R}^3$ defined by $y = \sqrt{3}x, z = 0$ and $y = -\sqrt{3}x, z = 0$ respectively. Let $S$ be the set of all points $P(x,y,z)$ such that there exists a unique plane $\Pi_1$ containing $P$ and $L_1$ , a unique plane $\Pi_2$ containing $P$ and $L_2$ , and $\Pi_1$ is perpendicular to $\Pi_2$ . Which of the following points does NOT belong to the set $S$ ?

Options

A.
$(1, \sqrt{5}, 1)$
B.
$(2, \sqrt{14}, -1)$
C.
$(1, \sqrt{3}, 0)$
D.
$(\sqrt{2}, 2\sqrt{2}, 1)$

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

Angle Bisector Direction|Collinearity vs Coplanarity|Parametric form pitfalls

Why This Question Is Tricky

Examiner Levers

Negative Thinking

Uses "NOT", "EXCEPT", or "INCORRECT" to flip the logic gate.

Difficulty Boost : +2

Logical Assertion

Tests the causal relationship between two true statements.

Difficulty Boost : +4

Recursive Logic

The answer to part A is required to solve part B (Step-wise failure test).

Difficulty Boost : +5

Common Mistakes

This question tests the candidate's ability to identify domain constraints in 3D locus problems.

The 'Lethal Collision' occurs when students blindly compute the dot product of the normal vectors ( $\vec{n}_1 \cdot \vec{n}_2 = 0$ ) to find the locus equation ( $y^2 - 3x^2 - 2z^2 = 0$ ), failing to account for the geometric constraint that a unique plane requires non-collinear points.

When the moving point $P$ lies on $L_1$ or $L_2$ , the position vector becomes collinear with the line's direction, resulting in a zero normal vector (the 3D equivalent of a division-by-zero singularity in slope calculations).

Thus, points on $L_1$ and $L_2$ must be excluded from the locus, trapping students who only check if the point satisfies the algebraic equation of the cone.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Vector Algebra - Class 12
Concept	Angle Bisector Direction
Difficulty	8
Cognitive Load	8
Target Solve Time	180 sec
Access	Free