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Question #40864

Difficulty: 8/10 | 🟢 Free

The Question

Let $S$ be the locus of a point $P(x,y,z)$ in 3D space such that the square of its distance from the origin is equal to $4(d_1 + d_2 + d_3) - 9$ , where $d_1, d_2,$ and $d_3$ are the perpendicular distances of $P$ from the $yz$ , $zx$ , and $xy$ planes, respectively. If $M$ is the maximum possible distance between any two points lying on $S$ , find the value of $M^2$ .

Options

A.
12
B.
48
C.
108
D.
192

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|Completing the Square in Quadratics|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to handle modulus sign dependencies within 3D distance constraints.

The trap lies in the definition of distance to coordinate planes, which inherently involves absolute values ( $|x|, |y|, |z|$ ).

Students commonly drop the modulus, incorrectly assuming the locus is a single sphere $(x-2)^2+(y-2)^2+(z-2)^2=3$ , leading to a maximum distance equal to its diameter ( $M^2=12$ ).

By completing the square with the modulus intact, the true locus is revealed as a symmetric set of 8 disjoint spheres across all octants, requiring the student to maximize the distance across the spatial diagonals.

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