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Question #40726

Difficulty: 8/10 | 🟢 Free

The Question

A point $P$ with position vector $\vec{p}$ moves in space such that $\vec{p} \times (\hat{i} + \hat{j} + \hat{k}) = \hat{i} - 2\hat{j} + \hat{k}$ . Let $M$ and $N$ be two points with position vectors $2\hat{i} + 2\hat{j} - \hat{k}$ and $10\hat{i} + 10\hat{j} - 5\hat{k}$ respectively. Let $Q$ be the orthogonal projection of $P$ onto the line passing through $M$ and $N$ . If the locus of $P$ is restricted such that $Q$ lies strictly on the line segment $MN$ , find the maximum value of $|\vec{p}|^2$ .

Options

A.
590
B.
434
C.
588
D.
225

Check Your Answer

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

Locus of a Moving Point|Orthogonal projection of a line|Parametric form pitfalls

Why This Question Is Tricky

Common Mistakes

This question tests the precise handling of parametric bounds under orthogonal projection constraints.

A common trap is assuming the parameter $t$ of the moving point shares the same domain as the length of the target segment (e.g., $t \in [0, 12]$ ), or ignoring the spatial offset between the lines.

By forcing candidates to map the scalar projection $s = (\vec{p} - \vec{m}) \cdot \hat{u}$ to the geometric bounds $0 \le s \le |MN|$ , the problem reveals that $t$ is shifted ( $t-2$ ).

Failing to establish this exact bounded domain for $t$ leads to incorrect maximization of the distance function.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Straight Lines - Class 11
Concept	Locus of a Moving Point
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free