Before You Begin

Question #40759

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1: \frac{x}{2} = \frac{y}{2} = \frac{z}{1}$ and $L_2: \frac{x}{-1} = \frac{y}{-2} = \frac{z}{-2}$ be two lines. Let $B$ be the line which is the acute angle bisector of $L_1$ and $L_2$ . A plane $\Pi$ contains the line $B$ and the point $P(1, -1, 1)$ . Let $L_3$ be the line passing through the point $Q(4, 5, 2)$ and perpendicular to the plane $\Pi$ . What is the shortest distance between the line $B$ and the line $L_3$ ?

Options

A.
$\frac{3}{\sqrt{17}}$
B.
$\frac{1}{\sqrt{3}}$
C.
$\sqrt{2}$
D.
$\frac{2}{\sqrt{17}}$

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

Angle Bisector Logic|Line lying in a plane|Unit Vector Normalization|Shortest distance between skew lines

Why This Question Is Tricky

Common Mistakes

This question tests the student's rigorous understanding of 3D geometry, specifically the acute/obtuse angle bisector conditions and shortest distance logic.

A lethal trap is set for students who blindly add the direction vectors ( $\vec{d_1} + \vec{d_2}$ ) without checking the dot product.

Because $\vec{d_1} \cdot \vec{d_2} < 0$ , adding them yields the obtuse bisector, which leads to a perfectly clean but incorrect plane and a shortest distance of $1/\sqrt{3}$ .

Additionally, students might misinterpret the shortest distance between the skew lines as the perpendicular distance from the point $Q$ to the plane $\Pi$ (yielding $\sqrt{2}$ ), failing to realize that the common perpendicular lies entirely within the plane $\Pi$ .

Quick Facts About This Question

Question Information

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