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Question #40862

Difficulty: 8/10 | 🟢 Free

The Question

A point $Q(x,y,z)$ moves on the sphere $x^2+y^2+z^2=4$ such that the ratio of its distance from the $xy$ -plane to its distance from the $z$ -axis is equal to $x$ . Let $S$ be the set of all points on the locus of $Q$ where the distance of $Q$ from the $xy$ -plane is maximized. Find the value of $\sum_{P \in S} (x_P^2 + y_P^2 + z_P^2)$ .

Options

A.
4
B.
8
C.
12
D.
16

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Specialist

Targeted practice for niche, high-difficulty concepts that frequently appear in JEE Advanced exams.

Concepts Behind This Question

Locus via Distance Constraints|Ratio of Division (k:1)|Symmetry Utilization

Why This Question Is Tricky

Common Mistakes

The question tests the student's ability to handle locus problems involving distance constraints while avoiding the 'Symmetry Utilization' trap.

When setting up the ratio $\frac{|z|}{\sqrt{x^2+y^2}} = x$ , students often square both sides immediately, obtaining $z^2 = x^2(x^2+y^2)$ .

This algebraic step artificially introduces symmetry with respect to the yz-plane, masking the geometric reality that the ratio of two non-negative distances must be non-negative, strictly requiring $x \ge 0$ .

Consequently, when finding the points of maximum distance, failing to apply this domain restriction leads to counting extraneous points with negative x-coordinates, doubling the final sum.

Furthermore, missing the absolute value on the z-distance halves the sum, perfectly separating conceptual errors into the given options.

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