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Question #41740

Difficulty: 8/10 | 🟢 Free

The Question

Let $L$ be a fixed line in space passing through the point $P(3, 1, 2)$ with direction ratios $(1, -1, 2)$ . The origin of the coordinate system is shifted to a variable point $O'(x, y, z)$ without rotating the axes, defining a new coordinate system $(X, Y, Z)$ . Let $\Pi'$ be the plane $X + Y + Z = 0$ in the new system. The traces of the planes $X=0$ and $Y=0$ on $\Pi'$ are denoted by $L'_1$ and $L'_2$ respectively. The orthogonal projection of the fixed line $L$ onto $\Pi'$ intersects $L'_1$ and $L'_2$ at points $V'_1$ and $V'_2$ . If the area of $\triangle O'V'_1V'_2$ is zero, then the locus of $O'$ in the original coordinate system is:

Options

A.
$3x - y - 2z = 4$
B.
$x + y - 2z = -4$
C.
$3x + y + 2z = 12$
D.
$x - 5y + 4z = 6$

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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

Line lying in a plane|Area of Triangle via Determinants

Why This Question Is Tricky

Common Mistakes

This problem tests the invariance of projected areas under coordinate shifts, acting as a gauge transformation.

Students often attempt to calculate the area using brute-force determinant transformations, getting lost in the shifting intercepts and direction-cosine projection algebra.

The trap lies in failing to recognize the geometric invariant: the area becomes zero precisely when the projected line passes through the new origin.

Geometrically, this means the normal to the projection plane through the new origin must intersect the original line.

Those who miss this spatial relationship will be bogged down by the 2D projection algebra, effectively punishing algorithmic application without spatial understanding.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line lying in a plane
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free