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Question #40857

Difficulty: 8/10 | 🟢 Free

The Question

The four complex numbers $z_1, z_2, z_3, z_4$ are the roots of the polynomial equation $z^4 - 4(w - 3 + 4i)z^3 + \alpha z^2 + \beta z + \gamma = 0$ where $w, \alpha, \beta, \gamma \in \mathbb{C}$ . These points represent the projections of the vertices of a tetrahedron onto the complex plane. Given that the sum of the squares of their distances from the origin is $64$ , find the maximum possible value of $|w - 11 + 19i|$ .

Options

A.
13
B.
17
C.
21
D.
25

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

Centroid of a Tetrahedron|Vieta’s Formulas in Complex Domain|Geometrical Interpretation of Modulus

Why This Question Is Tricky

Common Mistakes

This question tests the intersection of Vieta's formulas and the geometric interpretation of complex modulus.

A common trap is attempting to use the algebraic identity $\sum z_i^2 = (\sum z_i)^2 - 2\sum z_i z_j$ to utilize the unknown coefficient $\alpha$ .

However, the problem provides $\sum |z_i|^2$ , not $\sum z_i^2$ .

Students must recognize that the centroid $z_c$ of the roots minimizes the sum of squared distances, leading to the geometric identity $\sum |z_i - z_c|^2 = \sum |z_i|^2 - n|z_c|^2 \ge 0$ .

This establishes a hidden inequality bound on the centroid, forcing a transition from algebraic root relations to a geometric locus problem (maximizing the distance from a specific point to a bounded disk).

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Centroid of a Tetrahedron
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free