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Question #40855

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(6, 0, 0)$ , $B(0, 6, 0)$ , $C(0, 0, 6)$ , and $D(0, 0, 0)$ be the vertices of a tetrahedron. Let $P(x, y, z)$ be a variable point. The position vectors of $A, B$ , and $C$ relative to $P$ are $\vec{a}, \vec{b}$ , and $\vec{c}$ respectively. Let $V$ be the volume of the tetrahedron formed by the vectors $\vec{a}, \vec{b}$ , and $\vec{c}$ . Let $\vec{g}$ be the position vector of the centroid of the tetrahedron $ABCD$ relative to $P$ . If the locus of $P$ satisfying the equation $V + 8 \vec{g} \cdot (\hat{i} + \hat{j} + \hat{k}) + 24 = 0$ is a plane, then the perpendicular distance of this plane from the origin is:

Options

A.
$4\sqrt{3}$
B.
$\frac{16\sqrt{3}}{7}$
C.
$6\sqrt{3}$
D.
$2\sqrt{3}$

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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|Geometric Interpretation of Cross Product|Position Vector Origin Dependence|Modulus Sign Dependency

Why This Question Is Tricky

Common Mistakes

This problem tests the geometric interpretation of the scalar triple product and the centroid of a tetrahedron, while embedding a critical 'Modulus Sign Dependency' trap.

Students often forget that volume is strictly positive and drop the absolute value when expanding the scalar triple product $V = \frac{1}{6}|[\vec{a}, \vec{b}, \vec{c}]|$ .

If the modulus is ignored, the equation incorrectly simplifies to $18 - 3t = 4t - 30$ , yielding the false plane $x+y+z = 48/7$ and an incorrect distance of $16\sqrt{3}/7$ .

By requiring the variable origin $P$ to satisfy the locus, the problem forces students to rigorously evaluate both branches of the absolute value equation, rejecting the extraneous root.

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