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Question #41619

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ be the line given by the equations $x = y$ and $z = 0$ . Let $L_2$ be a variable line parallel to the $z$ -axis such that the shortest distance between $L_1$ and $L_2$ is $2$ . Let $M$ be the point of intersection of $L_2$ with the plane $P: x + y + z = 1$ . The coordinate axes are translated without rotation such that the new origin is at $O'(1, -1, 1)$ . In the new coordinate system, the locus of $M$ consists of two parallel lines, $C_1$ and $C_2$ . If $d_1$ and $d_2$ are the perpendicular distances from the new origin to $C_1$ and $C_2$ respectively, then the value of $d_1 d_2$ is:

Options

A.
2
B.
4
C.
8
D.
$4\sqrt{2}$

Check Your Answer

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

Line-Plane intersection logic|Shortest distance between skew lines|Locus Problems via Transformation|Shifting Origin

Why This Question Is Tricky

Common Mistakes

This problem tests the invariance of geometric properties under coordinate transformations, specifically targeting the 'Lethal Collision' where students fail to correctly re-parameterize locus conditions after a shift of origin.

A common trap is assuming the distance condition $|\alpha - \beta| = 2\sqrt{2}$ directly translates to $|X - Y| = 2\sqrt{2}$ in the new frame, ignoring the relative shift between the $x$ and $y$ coordinates.

This leads to the incorrect product of 4.

Furthermore, it elegantly combines skew lines, line-plane intersection, and the 3D geometry property that the distance from the origin to the intersection of two perpendicular planes (one passing through the origin) is simply the perpendicular distance to the other plane.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Line-Plane intersection logic
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free