Before You Begin

Question #40905

Difficulty: 8/10 | 🟢 Free

The Question

Let $\vec{a} = 4\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = \hat{i} + \hat{j} + \hat{k}$ . Let $B(\vec{b})$ be a variable point on the sphere $|\vec{r}| = 9$ . A point $P(\vec{p})$ lies on the line segment $AB$ such that $AP:PB = k:1$ ( $k>0$ ). For any position of $B$ such that $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar, the volume of the parallelepiped with coterminous edges $k\vec{p}, k\vec{b}, k\vec{c}$ is $16$ times the volume of the tetrahedron with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ . If the locus of $P$ is a sphere $S$ , find the maximum distance of a point on $S$ from the origin.

Options

A.
6
B.
8
C.
10
D.
12

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

Ratio of Division (k:1)|Scalar Triple Product Volume Logic|Determinant as Geometric Area/Volume

Why This Question Is Tricky

Common Mistakes

This problem tests the integration of 3D geometry loci with the properties of the Scalar Triple Product (STP).

The primary trap lies in the determinant scaling property: students frequently assume the volume scales linearly with the scalar multiple of the vectors, erroneously writing $[k\vec{p}, k\vec{b}, k\vec{c}] = k[\vec{p}, \vec{b}, \vec{c}]$ instead of the correct cubic scaling $k^3[\vec{p}, \vec{b}, \vec{c}]$ .

Furthermore, the problem exploits the distinction between the volume of a parallelepiped (STP) and a tetrahedron ( $\frac{1}{6}$ STP).

Failing to account for the $k^3$ factor or the $\frac{1}{6}$ multiplier leads to an incorrect polynomial for $k$ , preventing the resolution of the locus.

Quick Facts About This Question

Question Information

Subject	Mathematics
Chapter	Introduction to Three-dimensional Geometry - Class 11
Concept	Ratio of Division (k:1)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free