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Question #41728

Difficulty: 8/10 | 🟢 Free

The Question

A line $L$ passes through the point $P(2, 1, -1)$ and is parallel to the vector $2\hat{i} - \hat{j} + 2\hat{k}$ . A point $Q$ lies on the line $L$ such that its perpendicular distance from the plane $\Pi: x + 2y - 2z = 5$ is $4$ units. If the line segment $PQ$ does not intersect the plane $\Pi$ , then the length of $PQ$ is:

Options

A.
$\frac{33}{4}$
B.
$\frac{39}{4}$
C.
$\frac{11}{4}$
D.
$\frac{13}{4}$

Check Your Answer

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

Parametric form pitfalls|Distance from point to plane|The Modulus Trap

Why This Question Is Tricky

Common Mistakes

This question tests the 'Modulus Trap' in 3D geometry.

When finding the intersection or distance of a parametrically defined point to a plane, the absolute value in the distance formula yields two possible parameters (representing points on parallel planes on either side).

Students often arbitrarily pick the positive root or fail to check the geometric constraint.

The trap forces the student to evaluate the position of the points relative to the plane's half-spaces.

By requiring that the segment PQ does not intersect the plane, the student must verify that the algebraic sign of the plane's equation evaluated at P and Q is identical, thereby discarding the extraneous root.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Three-Dimensional Geometry - Class 12
Concept	Parametric form pitfalls
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free