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Question #41390

Difficulty: 8/10 | 🟢 Free

The Question

Let $L_1$ be the line given by $\frac{x-1}{1} = \frac{y-1}{2} = \frac{z-1}{2}$ and $L_2(t)$ be the family of lines given by $\frac{x - (3t - t^2/2)}{2} = \frac{y - 2|t-2|}{1} = \frac{z + 8}{-2}$ , where $t \in \mathbb{R}$ . If $d(t)$ represents the shortest distance between the lines $L_1$ and $L_2(t)$ , then the minimum value of $d(t)$ is:

Options

A.
$\frac{1}{3}$
B.
$\frac{8}{3}$
C.
$\frac{16}{3}$
D.
$\frac{4}{3}$

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

Geometry Constraint Modeling|Shortest distance between skew lines|Modulus Function Breakdown

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to model a geometric constraint (shortest distance between skew lines) into a single-variable function involving a modulus.

The 'Lethal Collision' or trap lies in the optimization phase.

Students typically rely on setting the derivative to zero to find minimums.

If they ignore the modulus definition or blindly differentiate the piecewise functions, they will find critical points at $t=1$ or $t=5$ , leading to incorrect distances of $8/3$ or $16/3$ .

The true global minimum lies at the non-differentiable corner point $t=2$ , forcing students to analyze the domain constraints and boundary points of the piecewise function carefully.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Application of Derivatives - Class 12
Concept	Geometry Constraint Modeling
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free