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Question #40861

Difficulty: 8/10 | 🟢 Free

The Question

Let $P(x, y, z)$ be a point in $\mathbb{R}^3$ . Define two complex numbers associated with $P$ as: $w_1 = (2x + y) + i(y + z)$ $w_2 = (2x - y) + i(y - z)$ Let $S$ be the locus of points $P$ such that $|w_1|^2 + |w_2|^2 = 16$ and $w_1 \bar{w_2} + \bar{w_1} w_2 = 0$ . Find the maximum distance between any two points on the locus $S$ .

Options

A.
$4\sqrt{2}$
B.
$2\sqrt{5}$
C.
$4$
D.
$8$

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 via Distance Constraints|Geometrical Interpretation of Modulus|Locus Problems via Transformation

Why This Question Is Tricky

Common Mistakes

This question tests the isomorphism between complex moduli constraints and 3D Euclidean geometry.

The primary trap lies in the coordinate transformation: students often assume that a constraint like $|w_1|^2 + |w_2|^2 = k$ maps to a sphere, failing to account for the non-orthogonal scaling in the transformation which distorts the locus into an ellipsoid.

Furthermore, even if they correctly identify the ellipsoid $4x^2 + 2y^2 + z^2 = 8$ , they might ignore the second orthogonality condition or incorrectly assume the maximum distance lies along the major axis of the ellipsoid ( $4\sqrt{2}$ ).

The true locus is restricted to planar slices ( $z = \pm 2x$ ), altering the maximum distance to $2\sqrt{5}$ .

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Locus via Distance Constraints
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free