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Question #40899

Difficulty: 8/10 | 🟢 Free

The Question

Let $A(1, 1, 1)$ , $B(2, -1, 3)$ , and $C(-1, 2, 2)$ be three vertices of a tetrahedron $ABCD$ . The centroid $G$ of the tetrahedron is constrained to lie on the line $L$ , which is the intersection of the planes $P_1: 3x - y + z = 4$ and $P_2: x + y + 3z = 12$ . Furthermore, the coordinates of the fourth vertex $D(x, y, z)$ must satisfy the inequality $\frac{x + y + 3z - 27}{3x - y + z - 6} \le 0$ Find the number of possible positions for the vertex $D$ .

Options

A.
0
B.
1
C.
2
D.
Infinite

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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|Line lying in a plane|Denominator Restriction

Why This Question Is Tricky

Common Mistakes

This question tests the concept of the centroid of a tetrahedron constrained to a line, combined with a 'Denominator Restriction' trap.

Students correctly deduce that the locus of the fourth vertex D is a line formed by the intersection of two new planes.

When substituting the locus equations into the given inequality, the expression evaluates to -1/0.

The trap relies on students incorrectly interpreting -1/0 as negative infinity, which trivially satisfies the less-than-or-equal-to zero condition, leading them to wrongly conclude there are 'Infinite' solutions.

In reality, the denominator restriction strictly excludes the entire locus from the domain, resulting in an empty set.

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