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Question #40742

Difficulty: 8/10 | 🟢 Free

The Question

Let $P_1$ be the plane containing the lines $L_1: \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3}$ and $L_2: \frac{x-2}{1} = \frac{y-3}{2} = \frac{z-5}{3}$ . Let $P_2$ be a plane parallel to $P_1$ such that the perpendicular distance between $P_1$ and $P_2$ is $2\sqrt{3}$ . A line $L_3: \frac{x - |a-2|}{1} = \frac{y - |a-5|}{2} = \frac{z - 1}{3}$ lies entirely in $P_2$ . Let $S$ be the set of all possible real values of $a$ . For each $a \in S$ , a point $Q_a$ is defined as $(a, a, a)$ . If $S = \{a_1, a_2\}$ , let $I_1$ and $I_2$ be the images of $Q_{a_1}$ and $Q_{a_2}$ respectively in the plane $P_1$ . The distance between $I_1$ and $I_2$ is:

Options

A.
$7\sqrt{3}$
B.
$14\sqrt{3}$
C.
$\frac{14\sqrt{3}}{3}$
D.
$21$

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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

Distance Between Parallel Lines|Image of a point in a plane|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to integrate 3D geometry with algebraic constraints (modulus) and geometric transformations (isometry).

The primary trap lies in the modulus equation $|a-2| + |a-5| = 7 \text{ or } -5$ .

Students often fail to reject the $-5$ case properly or make sign errors, leading to incorrect sets for $a$ (e.g., assuming $a = \pm 7$ ).

A secondary, crucial differentiator is the calculation of the distance between the images.

Brute-forcing the coordinates of the images $I_1$ and $I_2$ is computationally heavy and prone to arithmetic errors.

The advanced student will recognize that reflection is an isometry, meaning the distance between the images is identical to the distance between the original points $Q_0$ and $Q_7$ , bypassing the tedious image coordinate calculations entirely.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Distance Between Parallel Lines
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free